Mixed linear-quadratic nonrecurrence conjecture for minimal systems

Let $(X,T)$ and $(X,S)$ be minimal systems on the same compact space $X$. Mixed linear-quadratic nonrecurrence conjecture. 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

$$T^{n_i}x\to x,\qquad T^{2n_i}x\to x,\qquad S^{n_i^2}x\to x.$$

This is a stronger mixed-recurrence obstruction involving two iterates of $T$ and a quadratic iterate of $S$; the source presents it as open.