Cheltsov's conjecture on Kawamata decompositions of nodal Fano threefolds

Let $X$ be a nodal Fano threefold of Picard rank one. A Kawamata type decomposition means a semiorthogonal decomposition of the type considered in the paper for such a threefold.

Cheltsov's conjecture. The only nodal Fano threefolds of Picard rank one with Kawamata type decompositions are the quadric, $V_5$, and $V_{22}$.

This conjecture concerns the classification of nodal Fano threefolds admitting Kawamata type semiorthogonal decompositions. The paper establishes obstructions for broad classes of nodal threefolds, but the stated classification is presented as a suggestion and remains open in the supplied text.