Generalized Sylvester conjecture for odd multiply perfect numbers

Let $m\geq 2$, and let $n$ be a positive integer. Define the divisor-sum function by

$$\sigma(n)=\sum_{d\mid n}d.$$

A positive integer is $m$-perfect (or multiply perfect) if

$$\sigma(n)=mn.$$

Generalized Sylvester conjecture. There is no odd $m$-perfect number for $m\geq 2$.

For $m=2$, an $m$-perfect number is an ordinary perfect number, so this generalizes Sylvester's conjecture. The paper motivates the extension by observing that all known multiply perfect numbers are even; the existence of odd $m$-perfect numbers remains open.