Primality conjecture for integers of the form $am+1$

Let $n=am+1$, where $a$ and $m$ are positive integers, and let $p$ be the least prime divisor of $m$. Primality conjecture. If $a<p$ and $m\mid\phi(n)$, then $n$ is prime. This conjecture would generalize the preceding primality criterion from numbers of the form $ap^k+1$ to a broader class of integers. The source reports substantial experimental evidence but does not establish the converse or otherwise resolve the conjecture.