The piecewise hereditary characterization of orbit categories equal to cluster categories

Let $\Lambda$ be a finite-dimensional algebra with $\operatorname{gl.\!dim} \Lambda \leqslant 2$ satisfying the standing $\tau_2$-finiteness assumption. Let the orbit category be the triangulated orbit category associated with $\Lambda$, and let the cluster category be $\mathscr{C}_{\Lambda}$. An algebra is piecewise hereditary if it is derived equivalent to a hereditary algebra.

Piecewise hereditary characterization. The orbit category coincides with the cluster category for $\Lambda$ if and only if $\Lambda$ is piecewise hereditary.

The paper has already established the implication for hereditary algebras and investigates how far the orbit category is from the cluster category for broader classes of algebras. It proves that wide classes of non-piecewise hereditary algebras do not have coinciding orbit and cluster categories; the full if-and-only-if assertion is presented as the most ambitious hope and remains unresolved here.