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

Let $R$ be a ring and $T\in \mathrm{M}\mathrm{o}\mathrm{d}-R$ be a (non-classical) tilting module of finite projective dimension. Let $\mathcal T=({\mathcal T}^{\leq0}, {\mathcal T}^{\geq0})$bethe$t$-structureon$D®$generatedby$T$and${\mathcal D}=({\mathcal D}^{\leq0}, {\mathcal D}^{\geq0})$bethenatural$t$-structure\mathrm{.}Weshowthatthepair$(\mathcal D, \mathcal T)$ is right filterable in the sense of [FMT14], that is, for any $i\in \mathbb{Z}$ the intersection ${\mathcal{D}}^{\ge i}\cap {\mathcal{T}}^{\ge 0}$ is the co-aisle of a $t$-structure. As a consequence, the heart of $\mathcal{T}$ is derived equivalent to $\mathrm{M}\mathrm{o}\mathrm{d}-R$.