The no mysterious points conjecture for locally acyclic cluster algebras

Let $A$ be a locally acyclic cluster algebra and let $\mathcal{A}$ be its corresponding cluster variety. Let $\mathcal{D}(\mathcal{A})$ denote the deep locus and $\mathcal{S}(\mathcal{A})$ the stabilizer locus, consisting of points with nontrivial stabilizer under $\operatorname{Aut}(A)$. The no mysterious points conjecture.

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

Equivalently, every point in the deep locus has a nontrivial cluster-automorphism stabilizer. The reverse inclusion is known from the free action on cluster tori, while the converse is the conjectural part.