A smooth projective counterexample to Bondal-Polishchuk's conjecture

For a particular smooth projective (weak Fano) threefold $X$ we show that the braid group action on the set of full exceptional collections in $D^b(X)$ is not transitive. This provides a counterexample to a conjecture of Bondal and Polishchuk from 1993. The conjecture was first disproved by Chang, Haiden, and Schroll, who constructed a family of partially wrapped Fukaya categories for which the transitivity fails. However, no counterexample of the form $D^b(X)$ for $X$ a smooth projective variety was previously known.