Power stability of good sequences for multiple ergodic convergence

Let $(s_n)$ be a sequence of integers, and let $\boldsymbol{\text{good for }\ell\text{-convergence}}$ mean that the sequence has the corresponding multiple ergodic convergence property for every $\ell$-tuple of bounded functions. Power-stability conjecture. If $(s_n)$ is good for $\ell$-convergence for every $\ell\in\mathbb{N}$, then for every $k\in\mathbb{N}$, the sequence $(s_n^k)$ is good for $\ell$-convergence for every $\ell\in\mathbb{N}$. The conjecture is motivated by the known implication from good $2$-convergence of $(s_n)$ to single mean convergence of $(s_n^2)$; the general assertion remains open.