Pinsker reduction conjecture for polynomial ergodic averages along primes

Let $(X,\mathcal B,\mu,\mathbb Z^d)$ be a measure-preserving system generated by invertible transformations $T_1,\ldots,T_d$, and let $p_{i,j}(n)\in\mathbb Z[n]$ be the polynomials from Theorem 1. Write $P_{\mu}(\mathbb Z^d)$ for the Pinsker $\sigma$-algebra, and let $\mathbb P=\{a_0<a_1<\cdots<a_n<\cdots\}$ be the set of all prime numbers. Pinsker reduction conjecture. Under the assumptions of Theorem 1, for every $f_1,\ldots,f_m\in L^{\infty}(X,\mathcal B,\mu)$, the limit

$$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=1}^{m}f_j\bigl(T_1^{p_{1,j}(a_n)}\cdots T_d^{p_{d,j}(a_n)}x\bigr)$$

exists almost everywhere if and only if, for every $h_1,\ldots,h_m\in L^{\infty}(X,P_{\mu}(\mathbb Z^d),\mu)$, the limit

$$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=1}^{m}h_j\bigl(T_1^{p_{1,j}(a_n)}\cdots T_d^{p_{d,j}(a_n)}x\bigr)$$

exists almost everywhere. The conjecture proposes that almost-everywhere convergence of these prime polynomial averages is completely determined by the Pinsker factor. The preceding theorem establishes this reduction for the special case in which each polynomial iterate uses only one transformation; the general case remains open in the source.