The fixed-exponent projective-prime conjecture

Fix a prime integer $n\ge 3$, and let $p$ range over prime numbers. Define

$$m=1+p+p^2+\ldots+p^{n-1}.$$

The fixed-exponent projective-prime conjecture. For every fixed prime $n\ge 3$, there are infinitely many prime values of $m$. This is a special case of the broader projective-prime infinitude problem, and the source presents it as reasonable but unproved.