Nilpotent approximate subgroup conjecture for general linear groups

Let $\alpha$ be a symmetric subset of $GL(n,\mathbb{F}_p)$ containing $1$ and satisfying

$$|\alpha^3|\le K|\alpha|$$

for some $K\ge 1$. Folklore conjecture. There are subgroups $S\triangleright P$ of $GL(n,\mathbb{F}_p)$ such that $S/P$ is nilpotent, $P$ is contained in $\alpha^{c(n)}$, and $\alpha$ is covered by $K^{c(n)}$ cosets of $S$, where $c(n)$ depends only on $n$.

This is proposed as a stronger structural characterization of small-tripling subsets of $GL(n,\mathbb{F}_p)$. The preceding result gives a related conclusion with a soluble quotient and a coset of a perfect subgroup contained in $\alpha^3$; the nilpotent-quotient formulation remains conjectural.