Bergelson’s partition-regularity question for nice recurrence

A set $R\subseteq\mathbb{N}_{>0}$ is called a set of nice recurrence if, for every measure-preserving system $(X,\mu,T)$, every measurable set $A\subseteq X$, and every $\varepsilon>0$, there exists $n\in R$ such that $\mu(A\cap T^{-n}A)\geq\mu(A)^2-\varepsilon$. The problem asks whether, for every set of nice recurrence $R$ and every finite coloring $c:R\to\{1,\ldots,k\}$, there exists a color $i\in\{1,\ldots,k\}$ such that the monochromatic set $R_i=\{n\in R:c(n)=i\}$ is itself a set of nice recurrence.