Discriminant–semi-orthogonal decomposition multiplicity conjecture

Let $Z$ be the toric variety associated to a wall-crossing, and suppose its derived category has a semi-orthogonal decomposition

$$D^b(Z)=\langle D^b(Z_0),D^b(Z_0),\ldots,D^b(Z_k),D^b(Z_k)\rangle,$$

where each $D^b(Z_i)$ occurs some number of times. Let $C_W$ be the curve associated to the wall $W$, and let $\overline{\nabla}_i$ be the closure of the corresponding discriminant component. Multiplicity conjecture. The multiplicity of $D^b(Z_i)$ in this decomposition agrees with the intersection multiplicity of $\overline{\nabla}_i$ with $C_W$. The conjecture links categorical wall-crossing multiplicities with geometric discriminant intersection multiplicities. It is proved in the paper when the torus $T$ has rank $2$, while the general statement remains open.