Schur–Brauer version of van der Waerden's theorem for semimodules

Let $G=B_1\cup\dotsm\cup B_q$ be a finite partition of a semimodule $(G,\pmb{+})$ over a semiring $(R,+,\cdot)$. For a subset $F\subseteq R$, write $Fb=\{fb:f\in F\}$. Schur–Brauer version of van der Waerden's theorem. One of the sets $B_j$ has the property that, for every finite subset $F$ of $R$, there are elements $a\in G$ and $b\in B_j$ with $b\not=\boldsymbol{o}$ such that

$$a\pmb{+}Fb\subseteq B_j.$$

This is posed as an open question closely related to the paper's semimodule versions of van der Waerden's theorem and to Schur–Brauer-type partition results. The source does not state a resolution.