Polynomial-dimension abelian-structure conjecture for linear approximate groups

Let $\mathbb F$ be a field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group for some $K\geq1$. Polynomial-dimension abelian-structure conjecture. There exist polynomially bounded functions $M_1,M_2\colon\mathbb Z_{>0}\to\mathbb R$ such that some abelian subgroup $H\leq\operatorname{GL}_d(\mathbb F)$ satisfies

$$|A^2\cap H|\geq\frac{1}{(2K)^{M_1(d)}}|A^2|^{1/M_2(d)}.$$

This conjecture seeks to improve the paper's quasi-polynomial dependence on the dimension $d$ to polynomial dependence. It is presented as a possible strengthening and remains open.