Six-primes conjecture for orders of ${\rm PSL}_2(p)$

Consider the four cases (a)–(d) for a prime $p$, in which the relevant factors of $p-1$ and $p+1$ are required to be prime; equivalently, each case gives a triple of linear polynomials whose values must all be prime. Six-primes conjecture. There are infinitely many primes $p$ satisfying the conditions in each of cases (a)–(d), and consequently infinitely many groups ${\rm PSL}_2(p)$ whose order is a product of six primes. The Bateman–Horn numerical estimates provide strong heuristic evidence, but no proof is known.