The boundary-frozen saturation characterization for surface cluster algebras

Let $(S,M)$ be a marked surface and let $\mathcal{A}^{\mathrm{bd}}$ be the cluster algebra from $(S,M)$ with boundary frozen variables. For a triangulation $T$ and an arc $\gamma$, let $N^{\mathrm{bd}}(T,\gamma)$ denote the associated Newton polytope. Boundary-frozen saturation characterization. If $N^{\mathrm{bd}}(T,\gamma)$ is saturated for all $T,\gamma$, then $(S,M)$ is $\mathbf{P}$ or $\mathbf{P}^{\bullet}$. The conjecture is motivated by counterexamples for punctured tori and by the exceptional behavior of finite-type cluster algebras; the supplied status evidence indicates that the broader saturation assertion is disproved by counterexamples.