Multiple recurrence 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. Suppose there exist $m_0,n_0\in\mathbb Z$ such that $R_j(m_0,n_0)=1$ for $j=2,\ldots,\ell$, and either $R_1(m_0,n_0)=1$ or $(m_0,n_0)$ is a simple zero of $R_1$. For $A\in\mathcal X$ with $\mu(A)>0$, consider

$$\liminf_{N\to\infty}\mathbb E_{m,n\in[N]}\mu\bigl(A\cap T^{-1}_{R_1(m,n)}A\cap\cdots\cap T^{-1}_{R_\ell(m,n)}A\bigr).$$

Multiple recurrence conjecture. This liminf is positive. Furthermore, if all the rational polynomials have degree $0$ and $R_j(m_0,n_0)=1$ for $j=1,\ldots,\ell$ for some $m_0,n_0\in\mathbb Z$, then the conclusion holds for all multiplicative actions. The conjecture generalizes the known one-polynomial case, which follows from results cited in the source. It predicts multiple recurrence for substantially broader polynomial patterns, while the case of several polynomials remains open.