Derived equivalence conjecture for Calabi–Yau fibrations from homogeneous roof bundles

Let $G/P$ be a homogeneous roof, and $\mathcal{Z}$ a homogeneous roof bundle of type $G/P$ with projective bundle structures

$$p_i:\mathcal{Z}\longrightarrow\mathcal{Z}_i\qquad (i\in\{1,2\}).$$

Given a general section $S\in H^0(\mathcal{Z},\mathcal{L})$, define the Calabi–Yau fibrations $X_i:=Z(p_{i*}S)$. Derived equivalence conjecture. The Calabi–Yau fibrations $X_1$ and $X_2$ are derived equivalent:

$$D^b\operatorname{Coh}(X_1)\simeq D^b\operatorname{Coh}(X_2).$$

This conjecture extends the derived-equivalence phenomenon established for several specific roof types, including $A^M_n$, $A_n\times A_n$, $A^G_4$, $C_2$, and $G_2$. The cited mutation-based proofs verify the required conditions in those cases, while the assertion for arbitrary homogeneous roofs remains open.