The no mysterious points conjecture for skew-symmetric locally acyclic cluster algebras

Let $A$ be a skew-symmetric, locally acyclic cluster algebra in which all frozen variables are invertible, and let $\mathcal{A}=\operatorname{Spec}A$. Let $\mathcal{D}(\mathcal{A})$ be its deep locus and let $\mathcal{S}(\mathcal{A})$ be the locus of points with nontrivial stabilizer under the cluster automorphism group $\operatorname{Aut}(A)$. The no mysterious points conjecture.

$$\mathcal{D}(\mathcal{A})=\mathcal{S}(\mathcal{A}).$$

The equality says that every point of the deep locus is explained by an intrinsic cluster symmetry. The inclusion $\mathcal{S}(\mathcal{A})\subseteq\mathcal{D}(\mathcal{A})$ is immediate from the freeness of the automorphism action on cluster charts; the converse is conjectural in the stated generality.