  Enochs Conjecture asserts that each covering class of modules (over any ring)
has to be closed under direct limits. Although various special cases of the
conjecture have been verified, the conjecture remains open in its full
generality. In this paper, we prove the conjecture for the classes
$\mathrm{Filt}(\mathcal S)$ where $\mathcal S$ consists of $\aleph_n$-presented
modules for some fixed $n<\omega$. In particular, this applies to the left-hand
class of any cotorsion pair generated by a class of $\aleph_n$-presented
modules.
  Moreover, 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. This leaves us with no explicit example of a
covering class where we cannot prove that the Enochs Conjecture holds (possibly
under some additional set-theoretic assumption).
