Conjecture on rank-4 chiral polytopes for projective special linear groups

Let $G\cong PSL(2,q)$ with $q=p^d$ and $d>1$. A chiral polytope existence conjecture asserts that $G$ is the automorphism group of a chiral polytope of rank $4$ if and only if one of the following conditions on $q$ is satisfied:

$$4<q\equiv0\pmod{2};$$ $$9<q\equiv1\pmod{4};$$

or $q=p^d\equiv3\pmod{4}$, where $d>1$ is not a prime power. The problem concerns the remaining cases not settled by the preceding theorem; computations support the conjecture for many values, up to $q=11^{21}$, but no general proof is supplied.