Polynomial multiple ergodic averages conjecture

Let $T:\mathbb{Z}^D\curvearrowright (X,\mu)$ be a probability-preserving action, and let $\mathbf{p}_1,\mathbf{p}_2,\ldots,\mathbf{p}_k:\mathbb{Z}\to\mathbb{Z}^D$ be polynomial mappings. For $f_1,f_2,\ldots,f_k\in L^\infty(\mu)$, consider the averages

$$\frac{1}{N}\sum_{n=1}^N (f_1\circ T^{\mathbf{p}_1(n)})(f_2\circ T^{\mathbf{p}_2(n)})\cdots (f_k\circ T^{\mathbf{p}_k(n)}).$$

Polynomial multiple ergodic averages conjecture. These averages converge in $L^2(\mu)$ for every choice of $f_1,f_2,\ldots,f_k\in L^\infty(\mu)$. This would extend the known $L^2$-convergence theory for linear multiple ergodic averages to arbitrary polynomial mappings and general commuting actions; the conjecture remains open in general, although several special cases are known.