On a property of $t$-structures generated by non-classical tilting modules

Let $R$ be a ring and $T \in {\rm Mod-}R$ be a (non-classical) tilting module of finite projective dimension. Let $\mathcal T=({\mathcal T}^{\leq0}, {\mathcal T}^{\geq0})$ be the $t$-structure on $D(R)$ generated by $T$ and ${\mathcal D}=({\mathcal D}^{\leq0}, {\mathcal D}^{\geq0})$ be the natural $t$-structure. We show that the pair $(\mathcal D, \mathcal T)$ is right filterable in the sense of [FMT14], that is, for any $i\in\mathbb Z$ the intersection ${\mathcal D}^{\geq i}\cap {\mathcal T}^{\geq 0}$ is the co-aisle of a $t$-structure. As a consequence, the heart of $\mathcal T$ is derived equivalent to ${\rm Mod-}R$.