The intersection-density conjecture for \operatorname{PSL}_2$q$ on 3-sets

Let $q$ be a prime power, and let $\operatorname{PSL}_2(q)$ act on the $3$-sets of its natural degree-$q+1$ action. Intersection-density conjecture. If $q^2\equiv 1\pmod{5}$, then the intersection density is $4/3$; if $q^2\equiv 4\pmod{5}$, then the intersection density is $1$. These values would determine the intersection density in the two stated congruence cases.