Aymone's conjecture for sums over $k$-free integers

Let $k\geq 2$ be an integer, let $\mu^{(k)}$ denote the indicator function of the $k$-free integers, and let $h:\mathbb{N}\to\{-1,1\}$ be multiplicative. Aymone's conjecture. For every $\varepsilon>0$,

$$\sum_{n\leq x}\mu^{(k)}(n)h(n)=\Omega\left(x^{1/(2k)-\varepsilon}\right).$$

This is stated as a broader belief extending the squarefree case; the source gives no evidence that it has been resolved.