Extremal t-space-intersecting sets in finite general linear groups

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. A subset $Y\subseteq G_n$ is $t$-space-intersecting when every pair of elements of $Y$ is $t$-space-intersecting. For positive integers $t$ and sufficiently large $n$ compared to $t$, space-intersection conjecture. If $Y$ has size equal to the bound in Theorem, then $Y$ is a coset of the stabiliser of a $t$-space or a coset of the stabiliser of an $(n-t)$-space. This generalises the solved $t=1$ case attributed to Meagher and Spiga; the general assertion remains open in the supplied text.