The cubic-prime-power projective-prime conjecture

Let $p$ be a prime, set $q=p^3$, and define

$$m=1+q+q^2.$$

The cubic-prime-power projective-prime conjecture. There are infinitely many values of $q=p^3$ such that $m=1+q+q^2$ is prime. The conjecture is supported by an extensive computational search, but remains open.