Auto-equivalence and cluster automorphism group isomorphism conjecture

Let $\mathcal{C}$ be a $2$-Calabi-Yau triangulated category with cluster structure, let $T$ be a cluster tilting object, and let $\phi$ be a cluster map sending cluster tilting objects reachable from $T$ to clusters in the cluster algebra $\mathcal{A}_{\phi(T)}$. Let $Aut_T(\mathcal{C})$ be the quotient group of covariant and contravariant triangulated auto-equivalences that map $T$ to a reachable cluster tilting object, modulo equivalences agreeing on $T$. Auto-equivalence and cluster automorphism conjecture. There is a natural isomorphism

$$Aut_T(\mathcal{C})\cong Aut(\mathcal{A}_{\phi(T)}).$$

The conjecture is known when $\mathcal{C}$ is algebraic and the Gabriel quiver of $\operatorname{End}_{\mathcal{C}}(T)$ is acyclic. For generalized cluster categories, it is related to the conjecture that quivers determine potentials up to right equivalence; the general case remains open.