Polynomial Szemerédi-type conjecture for multiplicatively closed cut-and-project sets

Let $R$ be a locally compact and second countable ring, and let $\Lambda \subset R$ be a multiplicatively closed cut-and-project set. Suppose that $P_o \subset \Lambda$ has positive upper Banach density in $(R,+)$. Let $r\geq 1$, and let $p_1,\ldots,p_r:R\to R$ be polynomials satisfying $p_k(0)=0$ and $p_k(\Lambda)\subset \Lambda^q$ for some $q\geq 1$ and every $k$. For a finite set $F\subset \Lambda^\infty$, define

$$S_F=\{\lambda\in\Lambda:\text{there exists }\lambda_o\in P_o\text{ such that }\lambda_o+\{p_1(\lambda),\ldots,p_r(\lambda)\}\subset P_o\}.$$

Polynomial Szemerédi-type conjecture. The set $S_F$ is syndetic. This is proposed as a polynomial strengthening of the preceding corollary, motivated by polynomial extensions of Szemerédi's theorem; its resolution is not given in the supplied text.