Consider the following two symplectic matrices (given by the list of their lines)

A = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 1}, {0, 0, -1, 0}} and

B = {{-1, 0, 0, -1}, {0, 0, -1, 0}, {0, 1, -1, 0}, {1, 0, 0, 0}}

Is it true that the (Zariski-dense) group $<A,B>$ generated by A and B has infinite index in Sp(4,Z) (i.e., $<A,B>$ is thin in Sp(4,Z))?

This question emerged from some calculations performed by Vincent Delecroix and myself with the monodromy of certain square-tiled surfaces (and, in their turn, these calculations were motivated by a question posed by Peter Sarnak to Alex Eskin and Alex Wright).

More precisely, our considerations led to a representation $p:G \to Sp(4,Z)$ of the level 4 congruence group $G=<a,b>$ generated by the order three matrices a = {{0,-1},{1,-1}} and b = {{1,-3},{1,-2}} in SL(2,Z) such that p(a) = A and p(b) = B.

As it turns out, $G=<a>*<b>$ is the free product of two copies of Z/3Z (since {a,a^2} and {b,b^2} play ping-pong with some cones in R^2). Moreover, Vincent and I believe that the group <A,B> is thin because some numerical experiments with non-trivial words on A, A^2 and B, B^2 of length <25 seem to indicate that the representation p might be faithful (and, thus, $<A,B>$ would be thin as Sp(4,Z) doesn't contain finite-index subgroups isomorphic to free groups).

Nevertheless, after trying a couple of standard tricks (e.g., testing the injectivity of p on finite-index free subgroups of G or playing ping-pong in R^4, its exterior powers [and p-adic variants], etc.), Vincent and I are still unable to establish the thinness of <A,B> and/or the faithfulness of p, so that we would be thankful to any help with these problems!