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

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Let $Y,Z\subseteq G_n$ be $t$-space-cross-intersecting sets, meaning that every pair in $Y\times Z$ is $t$-space-intersecting. Cross-space-intersection conjecture. If their sizes meet the bound in Theorem, and $n$ is sufficiently large compared to $t$, then $Y=Z$ and $Y$ is the stabiliser of a $t$-space or the stabiliser of an $(n-t)$-space. This is the proposed equality classification for the cross-space-intersection bound; the supplied text gives no resolution.