The eigenspace-control conjecture for linear approximate groups

Let $\mathbb F$ be an algebraically closed field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group. Suppose every $a\in A^2$ has an eigenspace of dimension $(1-c)d$. The eigenspace-control conjecture. There exist subspaces $V_1\subset V_2\subset\mathbb F^d$ with

$$\dim V_2-\dim V_1>cd$$

such that at least $(2K)^{-O(d^{O(1)})}|A^2|$ elements $a\in A^2$ have some $\lambda\in\mathbb F$ satisfying $(a-\lambda)V_2\subset V_1$. This is proposed as a strengthening of the paper's regular-element proposition, removing its $\log\log d$ loss; it remains open.