The extended Dubrovin conjecture for Fano varieties

Let $X$ be a Fano variety of index $w$, meaning that $K_X$ is divisible by $w$ in $\operatorname{Pic} X$, and write $\omega_X={\mathcal O}(-w)$. Let $\mathrm{BQH}(X)$ and $\mathrm{QH}(X)$ denote the big and small quantum cohomology, and let

$$\mathrm{QH}(X)_{can}=\mathrm{QH}(X)\otimes_{\mathbb{Q}[q_1,\dots,q_s]}\mathbb{C},$$

where the specialization is induced by $K_X$. The class $[K_X]\in \operatorname{H}^2(X,\mathbb{C})\subset \mathrm{QH}(X)_{can}$ is considered with respect to quantum multiplication.

Extended Dubrovin conjecture. If $\mathrm{BQH}(X)$ is generically semisimple and $[K_X]$ is invertible in $\mathrm{QH}(X)_{can}$, then there is an exceptional collection $E_1,\dots,E_p$ extending to a full rectangular Lefschetz collection of $\mathbf{D}^{\mathrm{b}}(X)$, where

$$p=\frac{1}{w}\operatorname{dim}\operatorname{H}^{\bullet}(X,\mathbb{C}).$$

The conjecture refines the relation between quantum-cohomological semisimplicity and derived categories by incorporating Lefschetz decompositions. The source presents it as the version relevant to the paper; its general status is not stated as resolved.