The maximum-density conjecture for \operatorname{PSL}_2$q$ on the Kneser graph K$q+1,3$

Let $q$ be a prime power with $q\equiv 3\pmod{4}$, and consider the Kneser graph $K(q+1,3)$ and its automorphism group. Maximum-density conjecture. The group $\operatorname{PSL}_2(q)$ gives the maximum intersection density among all subgroups of the automorphism group of $K(q+1,3)$. Moreover, if $q^2\equiv 1\pmod{5}$, then the intersection density of $K(q+1,3)$ is $4/3$, while if $q^2\equiv 4\pmod{5}$, it is $1$. The claim would identify both an extremal group and the resulting density in the two congruence cases.