Filtrations of torsion classes in proper abelian subcategories

In an abelian category A, we can generate torsion pairs from tilting objects of projective dimension ≤ 1. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an associated torsion pair. Instead of trying to generate a torsion pair, Jensen, Madsen and Su generated a triple of extension closed classes that can filter any objects of A. We generalize this result to proper abelian subcategories.