Polynomial simultaneous nonrecurrence conjecture for minimal systems

Let $p$ and $q$ be integral polynomials vanishing at $0$, with $\deg p\ge 2$ and $\deg q\ge 2$. Let $(X,T)$ and $(X,S)$ be systems on the same compact space $X$. Polynomial simultaneous nonrecurrence conjecture. For any pair of such polynomials $p$ and $q$, there are minimal systems $(X,T)$ and $(X,S)$ such that for any $x\in X$ there is no subsequence $\{n_i\}$ of $\mathbb{Z}$, with $n_i\to\infty$, satisfying

$$T^{p(n_i)}x\to x\quad\text{and}\quad S^{q(n_i)}x\to x.$$

This proposes that simultaneous polynomial recurrence can fail at every point for suitably chosen minimal systems. The source presents it as open.