Results with FR4 1.6mm board underneath
Half-gap varied from 0.5 cm to 0.05 cm
Water layer at 0.25 cm above conductor
Total half-width 3.5 cm

Gap	Q at 1v
--> 0.5	1.328389067725905e-012
--> 0.45	1.35956769311746e-012
--> 0.4	1.394599144060286e-012
--> 0.35	1.433575714274607e-012
--> 0.3	1.477667882689791e-012
--> 0.25	1.528664682133727e-012
--> 0.2	1.591930100751743e-012
--> 0.15	1.675744199511223e-012
--> 0.09999999999999998	1.795229850953444e-012
--> 0.04999999999999999	2.010797859933693e-012


Now we switch water to air, and do all same as above

--> 0.5	4.793945198765056e-013
--> 0.45	5.020537761288981e-013
--> 0.4	5.285040508015764e-013
--> 0.35	5.595785079457721e-013
--> 0.3	5.964241450387937e-013
--> 0.25	6.407000203295059e-013
--> 0.2	6.981991300479198e-013
--> 0.15	7.769599906194133e-013
--> 0.09999999999999998	8.92787781464433e-013
--> 0.04999999999999999	1.106048796864962e-012

We calculate sensitivity as absolute and relative increase in Q:

0.5-Gap		Abs change	Rel change
0.5		8.49E-13	2.77E+00
0.45	8.58E-13	2.71E+00
0.4		8.66E-13	2.64E+00
0.35	8.74E-13	2.56E+00
0.3		8.81E-13	2.48E+00
0.25	8.88E-13	2.39E+00
0.2		8.94E-13	2.28E+00
0.15	8.99E-13	2.16E+00
0.1		9.02E-13	2.01E+00
0.05	9.05E-13	1.82E+00


We repeat the above process with water at 0.5 cm above conductor
Water
--> 0.5	8.752766590248886e-013
--> 0.45	9.009813142870823e-013
--> 0.4	9.301773355650984e-013
--> 0.35	9.638568925931499e-013
--> 0.3	1.003045297622213e-012
--> 0.25	1.049402593064754e-012
--> 0.2	1.108610802387971e-012
--> 0.15	1.188723101511406e-012
--> 0.09999999999999998	1.305682865274869e-012
--> 0.04999999999999999	1.51983929008905e-012

Air
--> 0.5	4.794455587822154e-013
--> 0.45	5.02079336494449e-013
--> 0.4	5.284328337342697e-013
--> 0.35	5.595129533265266e-013
--> 0.3	5.964203521593796e-013
--> 0.25	6.40739388777135e-013
--> 0.2	6.98291623610587e-013
--> 0.15	7.769642001765245e-013
--> 0.09999999999999998	8.929452491096684e-013
--> 0.04999999999999999	1.106452213049372e-012

0.5-Gap	Abs change	Rel change
0.5  3.96E-13	1.83E+00
0.45 3.99E-13	1.79E+00
0.4  4.02E-13	1.76E+00
0.35 4.04E-13	1.72E+00
0.3  4.07E-13	1.68E+00
0.25 4.09E-13	1.64E+00
0.2  4.10E-13	1.59E+00
0.15 4.12E-13	1.53E+00
0.1  4.13E-13	1.46E+00
0.05 4.13E-13	1.37E+00


Next we simulate the impact of total half-width with a 0.05 cm half-gap
And water at 0.25 cm above conductor
The charge Q at 1v is divided by half-width for a fair comparison

--> 3.5	5.745136742667702e-013
--> 3.2	6.005421911044168e-013
--> 2.9	6.300449304578417e-013
--> 2.6	6.656642868403633e-013
--> 2.3	7.081677217351463e-013
--> 2	7.629455466312921e-013
--> 1.7	8.313528640726256e-013
--> 1.4	9.28162978934003e-013
--> 1.1	1.071233563679295e-012
--> 0.8000000000000003	1.307958537478343e-012
--> 0.5	1.791828825493916e-012

Now with air
--> 3.5	3.160139419614243e-013
--> 3.2	3.434646505540171e-013
--> 2.9	3.754406201619345e-013
--> 2.6	4.146652117829625e-013
--> 2.3	4.622384012477374e-013
--> 2	5.240727563165074e-013
--> 1.7	6.01958835173584e-013
--> 1.4	7.121694476775139e-013
--> 1.1	8.744513986640323e-013
--> 0.8000000000000003	1.141425343361023e-012
--> 0.5	1.67764055657697e-012

This gives
0.5-width	Abs change	Rel change
3.5  2.58E-13	1.82E+00
3.2  2.57E-13	1.75E+00
2.9  2.55E-13	1.68E+00
2.6  2.51E-13	1.61E+00
2.3  2.46E-13	1.53E+00
2    2.39E-13	1.46E+00
1.7  2.29E-13	1.38E+00
1.4  2.16E-13	1.30E+00
1.1  1.97E-13	1.23E+00
0.8  1.67E-13	1.15E+00
0.5  1.14E-13	1.07E+00

Now the same with a 0.25 cm half-gap

Water
--> 3.5	4.367613377524871e-013
--> 3.2	4.502210606739919e-013
--> 2.9	4.645773588540636e-013
--> 2.6	4.817984220906396e-013
--> 2.3	5.008263007901083e-013
--> 2	5.230695052130344e-013
--> 1.7	5.500393007925514e-013
--> 1.4	5.854709610800016e-013
--> 1.1	6.346631550681427e-013
--> 0.8000000000000003	7.079790255432383e-013
--> 0.5	8.287838984944489e-013

Air
--> 3.5	1.83057148665575e-013
--> 3.2	1.984105951303681e-013
--> 2.9	2.157716723252501e-013
--> 2.6	2.372445420852333e-013
--> 2.3	2.622590676620939e-013
--> 2	2.925437557861282e-013
--> 1.7	3.30681298930799e-013
--> 1.4	3.81462715740756e-013
--> 1.1	4.531894917750372e-013
--> 0.8000000000000003	5.619685154157975e-013
--> 0.5	7.445086201571679e-013

0.5-Width	Abs change	Rel change
3.5		2.54E-13	2.39E+00
3.2		2.52E-13	2.27E+00
2.9		2.49E-13	2.15E+00
2.6		2.45E-13	2.03E+00
2.3		2.39E-13	1.91E+00
2  		2.31E-13	1.79E+00
1.7		2.19E-13	1.66E+00
1.4		2.04E-13	1.53E+00
1.1		1.81E-13	1.40E+00
0.8		1.46E-13	1.26E+00
0.5		8.43E-14	1.11E+00

Now, check what effect putting the FR4 board on top causes
We compare with the 0.05 cm half-gap and water at 0.25 cm above board
With conductor on top this gave:
Q=2.010797859933693e-012 in water
Q=1.106048796864962e-012 in air
Abs change	Rel change
9.05E-13	1.82E+00

Now with board on top
Q=1.94468e-012 in water
Q=1.1754e-012 in air
Abs change	Rel change
7.69E-13	1.65E+00

Compare this to the 0.5 cm height water above conductor with 0.05 cm half-gap:
4.13E-13	1.37E+00

That is to say, adding the 0.16 cm board on top does not make the performance worse than adding 0.25 cm air on top.
A thinner board like 0.04 cm would then probably be even better.
With 0.06 cm thickness,
Q=1.85455e-012 in water
Q=1.00265e-012 in air
Abs change	Rel change
8.52E-13	1.85E+00

This is comparable to the conductor on top with slightly reduced absolute change

Conclusions:
1. A smaller gap between conductor plate edges will increase the absolute change in capacitance that is sensed, but it will increase the base capacitance by a higher amount so relative change is lower. The improvement is not substantial however - under 10% increase in absolute change for a 90% gap reduction.
2. A wider plate width increases both the total and relative change in capacitance that is sensed. The relative change increase is linear, while absolute change slows down as width increases, meaning that with a wider sensor the contribution of base capacitance is less important. In terms of absolute capacitance, it seems that a width of about 40x gap size (2 cm half-width for 0.05 cm half-gap) is in the upper range of performance, while about 20x gap size (1 cm half-width for 0.05 cm half-gap) is towards the low end.
3. Placing the FR4 board on top of the conductor reduces the absolute and relative change compared to having the conductor on top, however this reduction is not as severe as an equivalent thickness of air. If using the FR4 board on top, it makes sense to have it as thin as possible.

Combining all this, given that absolute change will be readily measured, the increased absolute sensitivity of the smaller gap provides an advantage despite the increased base capacitance, so we can use a 1-mm gap between plates. The width of the plates suggested by optimization is to be as wide as feasible, and I think between getting adequate edge sensitivity and interleaved ground coverage, a width of about 1 inch is preferred. Note that a checkerboard pattern could be made (even with a one-sided board) to get good edge sensitivity but not sure if this is actually needed because adjacent pads can provide ground coverage. Finally, if a thin board is ordered, it would be possible to place the board facing upwards without much performance loss.

However due to interleaving pattern, there will be an increased length of zone between plates, so we need to decrease base capacitance as feasible. Using a 4 mm gap between plates will have about 1% reduction on measured absolute change, while reducing base capacitance by almost 50%. Further, with interleaved plates the width can be made wider, say 1.9 inches (so that 6x gives 11.4 inches), which will be better than 1 inch also in terms of lower base capacitance. Also, a thinner board is preferred even if not placing it on top, because it will be more flexible to glue to the underlying support.