An IP-star recurrence conjecture for semigroup homomorphisms

Let $S=(S,\cdot)$ and $R=(R,\cdot)$ be semigroups, with $R$ admitting a left invariant mean, and let $A\subseteq R$ be piecewise syndetic. For commuting homomorphisms $\varphi_1,\dots,\varphi_n:S\to R$, let $D(t^{-1}A;\varphi_1,\dots,\varphi_n)$ denote the set defined in the paper, and let an $\mathbf{IP}_r^*$ set in $S$ be a set meeting every $r$-term finite-products set. The conjecture. For every $n$, there exist $t\in R$ and $r\in\mathbb{N}$ such that

$$D(t^{-1}A;\varphi_1,\dots,\varphi_n)$$

is $\mathbf{IP}_r^*$ in $S$ for all commuting homomorphisms $\varphi_1,\dots,\varphi_n$ from $S$ to $R$. The conjecture seeks a noncommutative strengthening of the paper's recurrence results; the source notes that it would follow if a preceding theorem extended to arbitrary semigroups, but gives no proof or resolution.