EKR conjecture for PSL(2,q) on pairs of projective-line points

Let $q$ be a prime power, and let $\operatorname{PSL}(2,q)$ act on the $2$-subsets of the projective line $\operatorname{PG}(1,q)$. Recall that the group has the Erdős–Ko–Rado (EKR) property when its intersection density is $\rho(\operatorname{PSL}(2,q))=1$. EKR conjecture. If $q\equiv 1\pmod 4$, then $\rho(\operatorname{PSL}(2,q))=1$; if $q\equiv 3\pmod 4$, then $\operatorname{PSL}(2,q)$ does not have the EKR property. This prediction is based on computational results for the unresolved family of actions in line 13; the source does not state a resolution.