Equality of power indices for divisor voting systems of prime multiples

Let $n$ be an abundant number, and let $p$ and $m$ be primes satisfying

$$p,m>\sigma(n).$$

Prime-multiple power-index conjecture. The divisors of the divisor voting systems of $pn$ and $mn$ have the same Shapley–Shubik power indices and the same Banzhaf power indices.

This proposes that sufficiently large prime multiples of the same abundant number produce divisor voting systems with identical values for both power indices. The supplied text gives no evidence that the claim has been proved or refuted.