Aspinwall–Plesser–Wang semiorthogonal-decomposition conjecture for secondary-polytope edges

Let $A\subset N=\mathbb Z^d$ and $Q=\operatorname{conv}(A)$, and let $F$ be an edge of the secondary polytope. Let $Z_F$ be the associated space, let $n_{\Gamma,F}$ be the multiplicity defined from the leading discriminant term for each non-empty face $\Gamma$ of $Q$, and let $X_{\mathbb L_\Gamma}$ be the toric Deligne–Mumford stack associated with $\Gamma$. Aspinwall–Plesser–Wang decomposition conjecture. The category $D^b(Z_F)$ admits a semiorthogonal decomposition with $n_{\Gamma,F}$ components equivalent to $D^b(X_{\mathbb L_\Gamma})$ for each non-empty face $\Gamma$ of $Q$; moreover, for every non-empty face $\Gamma$ of $Q$, there exists a spherical functor

$$D^b(D_\Gamma)\to D^b(X)$$

for any toric Deligne–Mumford stack $X$ determined by a triangulation corresponding to a vertex of the secondary polytope. This refines the APW proposal by incorporating the multiplicities of discriminant components; the surrounding discussion presents it as conjectural evidence and notes that computing these multiplicities can be difficult.