Never mind, I completely misunderstood the question. Thank you!

What's wrong with taking $A$ and $B$ to be sets of nonzero multiples of some specific but rather general polynomials $f$ and $g$? These should be of size $|\mathbb F|-1$, and the restrictions to any line would be of the same size?

@Elliot I am not an expert -- maybe it could be singular.

I suggest looking at how the tangent subspaces $Lie(G)x$ and $T(x,X^R)$ intersect as $x$ varies in $X^R$. I would want to look at examples where the dimension of the intersection jumps at some $x$

Have you tried some examples -- maybe ${\rm PGL}(2,\mathbb C)$ acting on $n$ distinct points on $\mathbb P^1$ for some large $n$?

You can probably make $D$ with positive minors by having $D_{ij}$ grow extremely fast in the lexicographic order so that one of the terms in each Pfaffian dominates the rest of the terms.

If you have $A' = A + t D$, then Pfaffians of $A'$ will have as their leading terms as $t\to \infty$ the Pfaffians of $D$ times the appropriate power of $t$. So positivity for $A'$ follows from that of $D$.

I suppose that existence could be proved by induction on $n$; there might also be some explicit construction(s).

So by scaling $D= A'-A$ by a large integer, the problem reduces to finding a skew-symmetric matrix with positive integer entries above the diagonal such that all Pfaffians are positive. In fact, one can ignore the integrality and allow for arbitrary nonnegative entries (which can then be approximated by rationals and zeroes replaced by small rationals).

@Alexey Ustinov I presume that this means that there are formulas for the eigenvectors that involve sums of $1/k$. The LCM in A025529 must be an artifact of trying to get integer values for the eigenvectors.

I suppose you are not surprised that the proof is related to Adams operations as well :)

You need to resolve the singularities of the quotient to get the K3 surface.

Great, thank you! I have not looked at these.

Thank you, I will take a look at it. I saw that reference in Luke's book. Was under the impression that the idea was to approximate $\Gamma(s)^N$ by $\Gamma(Ns-i)$, will look in more detail.