Extremal t-intersecting sets in finite general linear groups

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. For positive integers $t$ and sufficiently large $n$ compared to $t$, let $Y\subseteq G_n$ be a $t$-intersecting set, meaning that every pair of elements of $Y$ is $t$-intersecting. Write $Y^T=\{y^T:y\in Y\}$. Extremal-set conjecture. If $Y$ has size equal to the bound in Theorem, then $Y$ or $Y^T$ is a $t$-coset. This would refine the proved extremal bound and span characterization by identifying all extremal sets; the corresponding assertion is stated for sufficiently large $n$ and remains open in the source.