Power transfer from mean $(k\ell)$-convergence to $\ell$-convergence

Let $(s_n)$ be a sequence of integers, and let $k,\ell\in\mathbb{N}$. Power-transfer conjecture. If $(s_n)$ is good for mean $(k\ell)$-convergence, then $(s_n^k)$ is good for $\ell$-convergence. This is presented as a more general statement that would imply the preceding power-stability conjecture. It is known when $\ell=1$, while the cases $\ell\geq 2$ remain open.