The bounded-size obstruction conjecture for boundary-frozen snake graphs

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 $G_{T,\gamma}$ be the associated snake graph and let $N^{\mathrm{bd}}(T,\gamma)$ be the associated Newton polytope. Bounded-size obstruction conjecture. There exists $B\in\mathbb{N}$ such that if $G_{T,\gamma}$ has more than $B$ squares, then $N^{\mathrm{bd}}(T,\gamma)$ is not saturated. This is presented as a stronger version of the preceding conjecture, motivated by the possibility that repeated labels in sufficiently large snake graphs force nonsaturation; its resolution is not specified in the supplied text.