Pointwise convergence conjecture for polynomial multilinear ergodic averages

Let $\mathbb X=(X,\mathcal B,\mu,\mathcal T)$ be a measure-preserving dynamical system, where $(X,\mathcal B,\mu)$ is a nontrivial $\sigma$-finite measure space and $\mathcal T=(T_1,\dots,T_d)$ is a commuting family of invertible measure-preserving transformations. Let $m,k,d\in\mathbb N$, and let $\mathcal P=(P_{1,1},\dots,P_{d,m})$ be a family of $k$-variate polynomials satisfying $P_{i,j}(\mathbb Z^k)\subseteq\mathbb Z^k$. For an $m$-tuple $f=(f_1,\dots,f_m)$, define

$$A^{\mathcal P}_N f(x)=\mathbb E_{n\in[N]^k}\prod_{j\in[m]}f_j\bigl(T_1^{P_{1,j}(n)}\cdots T_d^{P_{d,j}(n)}x\bigr).$$

Pointwise convergence conjecture. For every such system, polynomial family, and $m$-tuple $f=(f_1,\dots,f_m)$ with $f_j\in L^\infty(\mathbb X)$ for $j\in[m]$, the limit

$$\lim_{N\to\infty}A^{\mathcal P}_Nf(x)$$

exists for $\mu$-almost every $x$.

This is a pointwise convergence conjecture for multilinear polynomial ergodic averages and would extend the scope of the Calderón transference principle. The supplied text identifies it as a conjecture posed in an external reference, but gives no resolution evidence; its status is therefore recorded as open.