Bergelson–Leibman polynomial ergodic averages convergence conjecture

Let $(X,\mathcal{X},\mu)$ be a probability space, let $T_1,\ldots,T_\ell\colon X\to X$ be commuting, invertible measure-preserving transformations, let $f_1,\ldots,f_\ell\in L^\infty(\mu)$, and let $p_1,\ldots,p_\ell\in\mathbb{Z}[t]$. Bergelson–Leibman conjecture. The limit

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_1(T_1^{p_1(n)}x)\cdot\ldots\cdot f_\ell(T_\ell^{p_\ell(n)}x)$$

exists in $L^2(\mu)$. This is a general polynomial extension of multiple ergodic average convergence; the paper establishes several cases, while the full assertion is not resolved in the supplied context.