Bellow–Furstenberg pointwise convergence conjecture for multi-parameter polynomial averages

Let $d,k\in\mathbb{Z}_+$, let $(X,\mathcal B(X),\mu)$ be a probability measure space, and let $\mathcal{T}=(T_1,\ldots,T_d)$ be a family of invertible commuting measure-preserving transformations on $X$. Let $\mathcal{P}=(P_1,\ldots,P_d)\subset\mathbb{Z}[m_1,\ldots,m_k]$ satisfy $P_j(0)=0$ for every $j\in\{1,\ldots,d\}$. For $f\in L^{\infty}(X)$, define the multi-parameter polynomial averages by

$$A_{M;X,\mathcal{T}}^{\mathcal{P}}f(x)=A_{M_1,\ldots,M_k;X,T_1,\ldots,T_d}^{P_1,\ldots,P_d}f(x).$$

Bellow–Furstenberg conjecture. For $\mu$-almost every $x\in X$, these averages converge as $\min\{M_1,\ldots,M_k\}\to\infty$.

The case $d=k=1$ with arbitrary polynomial $P_1\in\mathbb{Z}[n]$ was the open Bellow–Furstenberg problem and was solved by Bourgain. The conjecture extends this pointwise convergence question to arbitrary multi-parameter polynomial averages and remains open in general.