Affine concentration from a uniform eigenspace condition

Determine, for each d, the smallest integer m such that every K-approximate group A⊂GL_d(F) over an algebraically closed field, whose elements all have an eigenspace of dimension at least m, has an affine subspace of Mat_d(F) containing at least (2K)^{-O(d^{O(1)})}|A^2| elements of A^2.

Background

The proof of the linear-group theorem only requires that a positive proportion of A^2 lie in a proper affine subspace, whereas Proposition 4.6 supplies a more detailed subgroup-control conclusion. This question asks for the weakest eigenspace lower bound that guarantees the affine concentration actually needed by the argument.

The authors note that the question is meaningful even for genuine finite groups, where Burnside’s theorem suggests a connection with stabilizers of nontrivial flags.