The projective space derangement graph conjecture for PGL(n+1,q)

Let $\mathbb{P}_q^n$ be projective $n$-space, with $n\geq 1$, and let $\Gamma_{\operatorname{PGL}(n+1,q)}$ be the derangement graph of $\operatorname{PGL}(n+1,q)$ acting on $\mathbb{P}_q^n$. An independent set is a set of vertices containing no adjacent pair, and a coset of the stabilizer of a point or hyperplane is a left or right coset of the corresponding setwise stabilizer. The projective space derangement graph conjecture. Any independent set of maximal size in $\Gamma_{\operatorname{PGL}(n+1,q)}$ is either the coset of the stabilizer of a point or the coset of the stabilizer of a hyperplane. The source notes that both types attain the maximal size, while for $n\geq 2$ they are not conjugate; no classification beyond these examples is given.