Enochs's conjecture on covering classes of modules

Let $\mathcal{X}$ be a class of modules. A class is covering if every module $M$ has an $\mathcal{X}$-cover, namely an $\mathcal{X}$-precover $f\colon X\to M$ such that every $g\in\operatorname{End}(X)$ is an automorphism whenever $f\circ g=f$. Enochs's conjecture. Every covering class of modules is closed under direct limits. Enochs's theorem establishes the converse implication: every precovering class closed under direct limits is covering. Whether every covering class is closed under direct limits remains open.