The Mersenne divisor conjecture for sums of powers

Let $M$ be a Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. Mersenne divisor conjecture. The integer $\sigma(M^{2h})$ is divisible by a non-Mersenne prime. The conjecture concerns the possible prime divisors of divisor sums associated with Mersenne primes; the source reports partial progress and identifies remaining cases, but does not state a resolution.