Mean convergence conjecture for multiplicative actions and linearly factorable rational polynomials

Let $(X,\mathcal X,\mu,T_n)$ be a finitely generated multiplicative action, and let $R_1,\ldots,R_\ell$ be rational polynomials that factor linearly. For $F_1,\ldots,F_\ell\in L^\infty(\mu)$, consider the averages

$$\mathbb E_{m,n\in[N]}T_{R_1(m,n)}F_1\cdots T_{R_\ell(m,n)}F_\ell.$$

Mean convergence conjecture. These averages converge in $L^2(\mu)$ as $N\to\infty$. Furthermore, if all the rational polynomials have degree $0$, then the conclusion holds for all multiplicative actions. The conjecture extends the mean-convergence results for multiplicative actions beyond pairwise independent linear forms. The degree-zero case is expected to hold without finite generation, while the general linearly factorable case remains open; the source notes that the corresponding result is verified when the polynomials are pairwise independent linear forms.