Kuznetsov–Smirnov's quantum-spectrum conjecture for Fano varieties

Let $X$ be a Fano variety of index $m$ over an algebraically closed field $\boldsymbol{\Bbbk}$ of characteristic zero, and assume that the big quantum cohomology $\operatorname{BQH}(X)$ is generically semisimple. Let $\operatorname{QS}^{\times}_X$ and $\operatorname{QS}^{\circ}_X$ denote the nonzero and zero parts, respectively, of the quantum spectrum, and let $\operatorname{Aut}(X)$ act on $X$ and its derived category. A rectangular Lefschetz collection has blocks indexed by twists $0,\ldots,m-1$, and its residual category is the orthogonal complement of these blocks. Kuznetsov–Smirnov's conjecture. There is an $\operatorname{Aut}(X)$-invariant exceptional collection $E_1,\dots,E_k$ in $\mathbf D^{\mathrm b}(X)$, where $k$ is the length of $\operatorname{QS}^{\times}_X$ divided by $m$, extending to the rectangular Lefschetz collection

$$E_1,\dots,E_k; E_1(1),\dots,E_k(1); \dots; E_1(m-1),\dots,E_k(m-1)$$

in $\mathbf D^{\mathrm b}(X)$. Its residual category $\mathcal R$ has a completely orthogonal $\operatorname{Aut}(X)$-invariant decomposition

$$\mathcal R=\bigoplus_{\xi\in\operatorname{QS}^{\circ}_X}\mathcal R_\xi,$$

where the components are indexed by closed points $\xi\in\operatorname{QS}^{\circ}_X$, and $\mathcal R_\xi$ is generated by an exceptional collection whose length equals the length of the localization $(\operatorname{QS}^{\circ}_X)_\xi$ at $\xi$. Finally, the induced polarization $\tau_{\mathcal R}$ sends $\mathcal R_\xi$ equivalently to $\mathcal R_{g(\xi)}$ for each $\xi$, where $g$ is a generator of $\mu_m$. These predictions relate the quantum cohomology spectrum of a Fano variety to its Lefschetz decomposition and residual category; their general validity is not established.