Cancellative elements in bend quotients of tropical principal prime ideals

Let $K$ be a field with valuation $K\to S$, where $S$ is a sub-semifield of $\mathbb{T}$, and let $\mathcal{M}$ be a toric monoid. Let $I\subseteq K[\mathcal{M}]$ be a principal prime ideal containing neither a monomial nor a binomial, and set

$$J=\operatorname{trop} I\subseteq S[\mathcal{M}].$$

Cancellative-elements conjecture. An element of $S[\mathcal{M}]/\operatorname{Bend}(J)$ is cancellative if and only if it is the image of a term of $S[\mathcal{M}]$.

This gives a concrete description of cancellative elements in the tropical coordinate semirings used to compute total semirings of fractions. The examples motivate the claim, but no proof or resolution is supplied here.