Independence of the cluster-model property for triangulated-surface quivers

Let $\mathbb{T}$ be a triangulation of a not necessarily orientable surface, and suppose that there is a rigid potential $W_l$ on the quiver-with-potential $\Diamond_l(\mathbb{T})$. Independence conjecture. Whether $(\Diamond_l(\mathbb{T}),W_l)$ is a cluster model does not depend on $l$. This asserts that the cluster-model property is invariant under the parameter $l$ for these triangulated-surface constructions; the supplied text gives no resolution.