Nilpotent obstruction conjecture for small tripling in linear groups

Let $K$ be a field, and let $A$ be a finite subset of $\operatorname{GL}_n(K)$ satisfying $A=A^{-1}$ and $e\in A$. For every $R\geq 1$, consider the alternatives involving the tripling set $A^3$. Nilpotent obstruction conjecture. Either

$$|A^3|\geq R|A|,$$

or there exist subgroups $H_1\leq H_2$ of $\operatorname{GL}_n(K)$ and an integer $k=O_n(1)$ such that $H_1$ and $H_2$ are both normal in $\langle A\rangle$, $H_2/H_1$ is nilpotent, $A^k$ contains $H_1$, and

$$|A^k\cap H_2|\geq R^{-O_n(1)}|A|.$$

This is posed as a broad question about describing subsets that fail to grow: failure of substantial tripling should be explained by concentration near a subgroup extension with nilpotent quotient. The supplied text gives no evidence that the proposed statement has been proved or disproved.