Popular-difference density conjecture for three-point matrix patterns in the integer lattice

Let $M_1$ and $M_2$ be $2\times2$ matrices with integer entries such that $M_2-M_1$ has full rank. For any $\alpha,\varepsilon>0$, there exists $N_0=N_0(\alpha,\varepsilon)\in\mathbb N$ such that, whenever $N\ge N_0$ and $A\subseteq\\{1,\dots,N\\}^2$ satisfies $|A|\ge\alpha N^2$, there is a vector $\vec n\in\mathbb Z^2$ with $M_1\vec n,M_2\vec n\ne0$ for which

\left|\left\\{\vec x\in\mathbb Z^2:\{\vec x,\vec x+M_1\vec n,\vec x+M_2\vec n\\}\subseteq A\right\\}\right|>(\alpha^3-\varepsilon)N^2.

Popular-difference density conjecture. The displayed lower bound holds for every pair of matrices satisfying the stated full-rank condition. The conjecture is a finitary form of the corresponding Khintchine-type recurrence question for $\mathbb Z^2$ and would determine a broad class of popular difference densities; the general three-point matrix-pattern problem is described as open.