Conjecture on polynomial patterns in relatively totally ergodic systems

Let $(X, \mu, T)$ be an ergodic $\mathbb{Z}$-system, let $\Phi$ be a Følner sequence, and let $a \in \operatorname{gen}(\mu, \Phi)$. Let $E \subseteq X$ be an open set with $\mu(E)>0$ satisfying

$$\mathbb{E}\left[\mathbbm{1}_E-\mu(E)\mid \mathcal{Z}_{rat}\right]=0.$$

Polynomial-pattern conjecture. For any integer-valued polynomial $P(x)\in\mathbb{Q}[x]$ such that both $P(x)$ and $P(x)+x$ are nonconstant, there exists $t\in\mathbb{Z}$ such that

$$\mathbf{EFS}^P_X(T^t a)\cap(E\times E)\ne\emptyset.$$

This conjecture predicts polynomial configurations in relatively totally ergodic systems, extending the rich polynomial-pattern consequences described in the surrounding discussion. Its resolution is not specified in the source.