Density conjecture for periodic derived orbit categories

Let $\Lambda$ be a finite-dimensional algebra, let $n$ be a positive integer, and write $\operatorname{\mathsf D^{\mathsf b}}(\operatorname{\mathsf{mod}} \Lambda)$ for the bounded derived category of finite-dimensional $\Lambda$-modules, $\Sigma$ for its suspension functor, and $\operatorname{\mathsf D_n}(\operatorname{\mathsf{mod}} \Lambda)$ for the triangulated hull of the orbit category by $\Sigma^n$. The canonical embedding is

$$\Delta \colon \operatorname{\mathsf D^{\mathsf b}}(\operatorname{\mathsf{mod}} \Lambda)/\Sigma^n \to \operatorname{\mathsf D_n}(\operatorname{\mathsf{mod}} \Lambda).$$

Density conjecture. For each positive integer $n$, the embedding $\Delta$ is dense if and only if $\Lambda$ is piecewise hereditary.

This conjecture characterizes when the periodic derived orbit category coincides with its triangulated hull. It is posed as the strongest expected form of the search for algebras whose orbit categories differ from their triangulated hulls; its resolution is not supplied in the source.