The bijection between cluster-tilting objects and d-angulations

Let $(S,M,D)$ be a $d$-angulation of a marked surface, let $(Q,W)$ be its associated quiver with superpotential, and let $\Gamma$ be the associated $d$-dimensional Ginzburg algebra. Define the generalized higher cluster category

$$\mathscr{C}_{\Gamma}:=\operatorname{per}(\Gamma)/\operatorname{pvd}(\Gamma)$$

and let $\pi:\operatorname{per}(\Gamma)\to\mathscr{C}_{\Gamma}$ be the projection functor. Let $C$ be the set of all $(d-2)$-cluster tilting objects obtained by finite mutations of $\pi(\Gamma)$, and let $C'$ be the set of all $d$-angulations obtained by finite flips of $(S,M,D)$. Bijection conjecture. There is a bijection between $C$ and $C'$ which sends $\pi(\Gamma)$ to $(S,M,D)$. This would identify finite mutations of the cluster-tilting object with finite flips of the associated $d$-angulation, extending the geometric model from the initial objects to their mutation and flip classes.