Enochs Conjecture for cotorsion pairs and more

Enochs Conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. In this paper, we prove the conjecture for the left hand-class of any cotorsion pair generated by (a class of) $\aleph_n$-presented modules for a fixed $n<\omega$. We also show that it is consistent with ZFC that Enochs Conjecture holds for all classes of the form $\mathrm{Filt}(\mathcal S)$ where $\mathcal S$ is a set of modules.