Simultaneous polynomial recurrence along prescribed residue classes

Let $(X,T^k)$ be minimal for some $k\geq 2$, let $d\in\mathbb N$, and let $P_m(n)$, $1\leq m\leq d$, be non-constant integral polynomials satisfying $P_m(0)=0$. Simultaneous polynomial recurrence conjecture. For every $0\leq j<k$, there is a sequence $\\{n_i\\}_{i\in\mathbb N}$ and a dense $G_\delta$ set of points $x\in X$ such that $n_i\equiv j\pmod{k}$ and

$$T^{P_1(n_i)}x\longrightarrow x,\ldots,T^{P_d(n_i)}x\longrightarrow x \qquad (i\longrightarrow\infty).$$

This strengthens the single-polynomial and quadratic recurrence results by requiring simultaneous convergence for several polynomials and a prescribed congruence class. The source presents it as requiring substantial further work, and gives no resolution.