Polynomial recurrence for totally minimal systems

Let $(X,T)$ be a totally minimal system, and let $P(n)$ be a non-constant integral polynomial. Polynomial recurrence conjecture. There is a dense $G_\delta$ subset $\Omega$ of $X$ such that, for every $x\in\Omega$, the set

$$\\{T^{P(n)}(x):n\in\mathbb Z\\}$$

is dense in $X$. The conjecture extends the quadratic polynomial case proved in the source and is also known for minimal weakly mixing systems. Its general case remains open.