Popular differences for matrix patterns

The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ a finite abelian group order $N$, $A\subseteq G^k$ with $|A|\ge\alpha N^k$. If $M_2$, $M_1-M_2$, $M_1+M_2$ are automorphisms $G^k$, is it true that there exists popular difference $d \in G^k\setminus\{0\}$ such \[\#\{x G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d A\} \ge (\alpha^4-o(1))N^k.\] We show this false in general, but holds for $G = \mathbb{F}_p^n$ $p$ an odd prime given the additional spectral condition no pair eigenvalues $M_1M_2^{-1}$ (over $\overline{\mathbb{F}}_p$) negatives each other. In particular, "rotated squares" pattern does not satisfy eigenvalue condition, we give construction set positive density $(\mathbb{F}_5^n)^2$ which has nonzero difference. This surprising contrast to three-point patterns, handle over all compact groups do require condition.