Powerfree preservation conjecture for the sum-of-proper-divisors function

Let $s(n):=\sum_{d\mid n,\ d<n}d$ be the sum-of-proper-divisors function. Fix $k\ge 2$, and call a positive integer $k$-free if it is not divisible by the $k$th power of any integer greater than $1$.

Powerfree preservation conjecture. On a set of integers $n$ of asymptotic density $1$,

$$\text{$n$ is $k$-free} \Longleftrightarrow \text{$s(n)$ is $k$-free}.$$

The conjecture predicts that, for every fixed $k\ge 2$, the sum-of-proper-divisors function preserves $k$-freeness for almost all integers. The preceding discussion explains that small prime-power divisors are controlled by results on the divisibility of $\sigma(n)$; the remaining issue is to exclude large prime-power divisors of $s(n)$ for almost all $n$.