Bergelson–Leibman conjecture on polynomial ergodic averages for nilpotent actions

Let $G$ be a nilpotent group of measure-preserving transformations of a probability space $(X,\mathcal B,\mu)$. For $T_1,\dots,T_l\in G$, $f_1,\dots,f_d\in L^\infty(X)$, and integer-valued polynomials $p_{j,i}$, consider the averages

$$\frac{1}{N}\sum_{n=1}^N\prod_{j=1}^d f_j\bigl(T_1^{p_{j,1}(n)}\cdots T_l^{p_{j,l}(n)}x\bigr).$$

Bergelson–Leibman conjecture. The limit of these averages exists in $L^2$-norm and almost everywhere. The conjecture concerns pointwise and norm convergence of polynomial multiple ergodic averages for nilpotent group actions; the paper presents it as a conjecture and does not indicate a resolution.