Let $X$ be a smooth complex projective variety and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. Then the braid group on $\mathrm{dim}\:K_0(X)$ strands acts by mutations on on the set of full exceptional collections in $D^b(X)$.

This action has been shown to be transitive for Fano threefolds with $b_2=1$ and $b_3=0$ (Polishchuk) or for 3-block collections on smooth del Pezzos (Karpov&Nogin).

Does there exist a smooth projective varietysuch that the action of braid group on the set of full exceptional collections is intransitive?