Gall-Rahav conjecture on non-Mersenne divisors of cyclotomic polynomial values

Let $h\in\mathbb{N}^*$ and let $M$ be a Mersenne prime, meaning a prime polynomial of the form $1+x^a(x+1)^b$. Assume that

$$M\notin\{1+x+x^3,1+x^2+x^3\}\quad\text{or}\quad h\geq 2.$$

Gall-Rahav conjecture. The polynomial $\sigma(M^{2h})$ is divisible by a non-Mersenne prime.

This conjecture concerns the factorization of values of cyclotomic-type sums associated with Mersenne primes. The paper proves a substantial partial result: for every prime $p\geq 5$, the corresponding polynomial $\phi_p(M)=\sigma(M^{p-1})$ has a non-Mersenne prime divisor when $M$ lies outside the explicitly defined exceptional sets, while the displayed general assertion is not resolved here.