Derived exchange-pair conjecture for cluster endomorphism algebras

Let $H$ be a finite-dimensional hereditary algebra, let $\overline{T}$ be an almost complete basic tilting object of the cluster category $\operatorname{\mathcal C}$, and let $M$ and $M^{\ast}$ be its two complements. Set

$$T=\overline{T}\coprod M,\qquad T'=\overline{T}\coprod M^{\ast},$$

and

$$\Gamma=\operatorname{End}_{\operatorname{\mathcal C}}(T)^{\operatorname{op}},\qquad \Gamma'=\operatorname{End}_{\operatorname{\mathcal C}}(T')^{\operatorname{op}}.$$

Let $S_M$ and $S_{M^{\ast}}$ be the simple tops of the corresponding Hom modules.

Derived exchange-pair conjecture. The quotient module categories are equivalent:

$$\operatorname{mod}\Gamma/\operatorname{add}S_M\simeq \operatorname{mod}\Gamma'/\operatorname{add}S_{M^{\ast}}.$$

This predicts a categorical relationship between the endomorphism algebras of the two completions of an almost complete tilting object; the supplied text gives no evidence of a resolution.