Geometricity conjecture for homomorphisms from Laurent-generated semirings

Let $A$ and $B$ be finite sets, and let $R$ be a Laurent-generated subsemiring of $\mathbb{Z}^A_{\mathrm{pos}}\cup\{-\infty\}$. A semiring homomorphism $\nu:R\to\mathbb{Z}^B_{\mathrm{pos}}\cup\{-\infty\}$ is called geometric when it is geometric with respect to some Laurent-generating set of $R$.

Geometricity conjecture. Any semiring homomorphism $\nu:R\to\mathbb{Z}^B_{\mathrm{pos}}\cup\{-\infty\}$ is geometric.

This is the general algebraic form of the geometricity question used for Boolean function semirings. The supplied text does not provide evidence that the conjecture has been solved or refuted.