Erdős–Ko–Rado conjecture for k-intersecting invertible matrices

Let $q$ be a prime power, let $k\in\mathbb{N}$, and let $\Gamma=\operatorname{GL}(n,\mathbb{F}_q)$. Define

$$X_{q,n,k}=\{A\in\operatorname{GL}(n,\mathbb{F}_q):\operatorname{rank}(A-I)>n-k\},\qquad G_{q,n,k}=\operatorname{Cay}(\Gamma,X_{q,n,k}).$$

The graph's independent sets are the $k$-intersecting families of invertible matrices, and $\alpha(G)$ and $\vartheta(G)$ denote its independence number and Lovász theta-number. Erdős–Ko–Rado conjecture. For each $q,k\in\mathbb{N}$, there exists $n_0=n_0(q,k)\in\mathbb{N}$ such that for all $n\geq n_0$,

$$\vartheta(G_{q,n,k})=\alpha(G_{q,n,k})=\prod_{i=k}^{n-1}(q^n-q^i).$$

The product is achieved by matrices fixing pointwise a chosen $k$-dimensional subspace. Computational evidence supports the assertion for small values of $n$ and $q$, but the stated asymptotic result remains open.