Bergelson–Leibman–Lesigne conjecture on jointly intersective polynomial recurrence

Let $p_1,\ldots,p_\ell\in\mathbb{Z}[t]$ be jointly intersective, meaning that for every $r\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that $r$ divides each of $p_1(n),\ldots,p_\ell(n)$. Let $(X,\mathcal{X},\mu,T_1,\ldots,T_\ell)$ be a system, meaning a standard probability space with commuting invertible measure-preserving transformations, and let $A\in\mathcal{X}$ satisfy $\mu(A)>0$. Bergelson–Leibman–Lesigne conjecture.

$$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu\big(A\cap T_1^{-p_1(n)}A\cap\cdots\cap T_\ell^{-p_\ell(n)}A\big)>0.$$

The conjecture seeks a complete characterization of polynomial families that are good for multiple recurrence; the jointly intersective condition is necessary because non-intersective polynomials admit positive-density sets avoiding the corresponding patterns. The source cites Bergelson, Leibman, and Lesigne for the conjecture and presents it as an open problem.