Ergodic polynomial odd-recurrence conjecture

Let $(X,\mu,T)$ be an ergodic measure-preserving system and let $k,d\in\mathbb{N}$. Suppose that $(X,\mu,T^k)$ is ergodic. For polynomials $p_1,\ldots,p_d$ over $\mathbb{Z}$ satisfying $p_i(0)=0$ for $i=1,\ldots,d$, and a measurable set $A\subseteq X$ with $\mu(A)>0$, define

$$R=\{n\in\mathbb{Z}:\mu(T^{-p_1(n)}A\cap\cdots\cap T^{-p_d(n)}A)>0\}.$$

Ergodic polynomial odd-recurrence conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

The source states that this is an ergodic-theoretic analogue equivalent to the topological polynomial odd-recurrence conjecture. No resolution is given in the supplied text.