Zero-entropy extension of the multiple ergodic averages theorem

Let $(X,\mathscr{B},\mu)$ be a probability space and let $T,R,S:X\rightarrow X$ be measure-preserving automorphisms. Suppose that $R$ and $S$ commute, $S$ is weakly mixing, and let $\ell\in\mathbb{N}$ and $p_1,\ldots,p_\ell\in\mathbb{Q}[x]$ be pairwise essentially distinct integer polynomials, each of degree at least $2$. For $f,h,g_1,\ldots,g_\ell\in L^\infty(X,\mu)$, with $\int_Xg_j\,d\mu=0$ for some $1\leq j\leq\ell$, consider the corresponding polynomial multiple ergodic average. Zero-entropy conjecture. Theorem 1.7 remains true when $T$ has zero entropy, rather than singular spectrum. This would substantially extend the theorem beyond the singular-spectrum hypothesis; the source presents it as an open conjectural extension motivated by the vanishing of most multiple correlations in the proof.