The Kuznetsov decomposition conjecture for Fano complete intersections

Let $X\subset\mathbf{P}^n$ be a smooth Fano complete intersection of degree $d\leq n$. Its Kuznetsov decomposition is

$$\operatorname{D^b_{coh}}(X)=\left\langle\mathrm{Ku}(X),\langle\mathcal{O}_X,\ldots,\mathcal{O}_X(n-d)\rangle\right\rangle,$$

where $\mathrm{Ku}(X):=\langle\mathcal{O}_X,\ldots,\mathcal{O}_X(n-d)\rangle^\perp$ is the Kuznetsov, or residual, component. A refinement and mutation of this decomposition means that the proposed decomposition is obtained by refining components and applying semiorthogonal mutations.

The Kuznetsov decomposition conjecture. For a Fano complete intersection $X\subset\mathbf{P}^n$, there exist $\tau\in\mathrm{H}^\bullet(X)$ and a sector $\mathscr{S}$ such that the preceding path-and-decomposition conjecture holds and the induced semiorthogonal decomposition is a refinement and mutation of the Kuznetsov decomposition.

This conjecture predicts that the noncommutative minimal model construction recovers the standard semiorthogonal decompositions known for Fano complete intersections, up to refinement and mutation. The existence of the required path and the claimed relationship remain open.