Theta-number conjecture for 1-intersecting invertible matrices

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

$$G_{q,n,1}=\operatorname{Cay}(\Gamma,X_{q,n,1}),\qquad X_{q,n,1}=\{A\in\operatorname{GL}(n,\mathbb{F}_q):\operatorname{rank}(A-I)>n-1\}.$$

Here $\alpha(G)$ denotes the independence number and $\vartheta(G)$ the Lovász theta-number. Theta-number conjecture. For all values of $n$ and $q$,

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

The product is attained by the family of invertible matrices fixing a chosen nonzero vector. The equality was verified numerically for small values of $n$ and $q$, but the assertion for all parameters remains open.