Density criterion for piecewise hereditary algebras

Let $\Lambda$ be a $\tau_2$-finite algebra of global dimension at most $2$. Let $\mathcal{D}^{\rm b}(\Lambda)$ be the bounded derived category of $\Lambda$-modules, let $\mathcal{C}_2(\Lambda)$ be the associated generalized cluster category, and let $\Pi_3(\Lambda)$ be the associated $3$-preprojective algebra. Density criterion for piecewise hereditary algebras. The algebra $\Lambda$ is piecewise hereditary if and only if the functor

$$\mathcal{D}^{\rm b}(\Lambda)\longrightarrow\mathcal{C}_2(\Lambda)$$

is dense. Equivalently, $\Lambda$ is piecewise hereditary if and only if every $\Pi_3(\Lambda)$-module is gradable. This conjecture gives a criterion for detecting piecewise hereditary algebras through the density of the canonical functor to the generalized cluster category, or equivalently through gradability of modules over the associated preprojective algebra. The source does not provide evidence of a resolution.