The intersection-density conjecture for \operatorname{PGL}_2$q$ on 2-sets

Let $q$ be an odd prime power, and let $\operatorname{PGL}_2(q)$ act on the $2$-sets of $\{1,\dots,q+1\}$. Intersection-density conjecture. The action has intersection density $1$. The source reports that computations support this for $q\leq 27$.