Averaging and recurrence conjecture for zero-entropy systems along superpolynomial times

Let $(X,\mathcal{B},\mu)$ be a probability space, and let $T,S:X\to X$ be two not necessarily commuting measure-preserving transformations. Let $0<c<1/2$, and suppose that $(X,\mu,T)$ has zero entropy. Write $\mathcal{I}_T$ and $\mathcal{I}_S$ for the $\sigma$-algebras of $T$-invariant and $S$-invariant sets, respectively. Averaging and recurrence conjecture. For every $f,g\in L^{\infty}(\mu)$,

$$\lim_{N\to\infty}\left\|\operatorname*{\mathbb{E}}_{n\in[N]}T^nf\,S^{\lfloor G(n)\rfloor}g-\operatorname*{\mathbb{E}}[f\mid\mathcal{I}_T]\operatorname*{\mathbb{E}}[g\mid\mathcal{I}_S]\right\|_2=0,$$

where the limit is in $L^2(\mu)$; and for every $A\in\mathcal{B}$,

$$\lim_{N\to\infty}\operatorname*{\mathbb{E}}_{n\in[N]}\mu\bigl(A\cap T^{-n}A\cap T^{-\lfloor G(n)\rfloor}A\bigr)\geq\mu(A)^3.$$

These conclusions are presented as a likely consequence of the preceding pointwise Furstenberg-system conjecture, so their status is conditional and not established in the supplied text.