The 2-Mersenne divisor conjecture

Let $S$ be a 2-Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. 2-Mersenne divisor conjecture. The integer $\sigma(S^{2h})$ is divisible by a non-Mersenne prime or by a non-2-Mersenne prime. The source presents this as one of two proposed conjectures concerning possible divisors of perfect polynomials; it gives motivation and conditional implications, but no resolution.