Glasner–Huang–Shao–Weiss–Ye polynomial odd-recurrence conjecture

Let $(X,T)$ be a minimal system and let $k,d\in\mathbb{N}$. Suppose that $(X,T^k)$ is minimal. For polynomials $p_1,\ldots,p_d\in\mathbb{Z}[x]$ satisfying $p_i(0)=0$ for $i=1,\ldots,d$, and a nonempty open set $U\subseteq X$, define

$$R=\{n\in\mathbb{Z}:T^{-p_1(n)}U\cap\cdots\cap T^{-p_d(n)}U\neq\emptyset\}.$$

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

This conjecture generalizes polynomial recurrence in minimal systems and is equivalent in the paper to a conjecture describing connected orbit closures in totally minimal nilsystems. Its resolution is not supplied here.