A derived equivalence between cluster equivalent algebras

Let $Q$ be an acyclic quiver. Associated with any element $w$ of the Coxeter group of $Q$, triangulated categories $\underset{‾}{\text{Sub}}{\mathrm{\Lambda }}_w$ were introduced in \cite{Bua2}. There are shown to be triangle equivalent to generalized cluster categories ${\text{Cc}}_{{\mathrm{\Gamma }}_w}$ associated to algebras ${\mathrm{\Gamma }}_w$ of global dimension $\le 2$ in \cite{ART}. For $w$ satisfying a certain property, called co-$c$-sortable, other algebras $A_w$ of global dimension $\le 2$ are constructed in \cite{AIRT} with a triangle equivalence ${\text{Cc}}_{A_w}\simeq \underset{‾}{\text{Sub}}{\mathrm{\Lambda }}_w$. The main result of this paper is to prove that the algebras ${\mathrm{\Gamma }}_w$ and $A_w$ are derived equivalent when $w$ is co-$c$-sortable. The proof uses the 2-APR-tilting theory introduced in \cite{IO}.