Prime-graph-complement invariance conjecture for PSL(2, q)-solvable groups

Let $p$ and $q$ be prime powers such that $\operatorname{PSL}(2,q)$ and $\operatorname{PSL}(2,p)$ are $K_4$-groups. Suppose one of the following holds: $p$ and $q$ are primes with $p,q\equiv 1\pmod{12}$; $p$ and $q$ are primes with $p,q\not\equiv 1\pmod{12}$; $p=2^f$ and $q=2^g$ for primes $f,g\geq 5$; or $p=3^f$ and $q=3^g$ with $f,g\neq 4$. For a graph $\Xi$, prime-graph-complement invariance conjecture.

$$\Xi\cong\operatorname{\overline{\Gamma}}(G)\text{ for some }G\text{ a }\operatorname{PSL}(2,q)\text{-solvable group}$$

if and only if

$$\Xi\cong\operatorname{\overline{\Gamma}}(H)\text{ for some }H\text{ a }\operatorname{PSL}(2,p)\text{-solvable group}.$$

This conjecture would help classify prime graph complements of $\operatorname{PSL}(2,q)$-solvable groups by transferring realizability between the specified parameters $p$ and $q$.