The Erdős–Goodman–Pósa conjecture for graphs with independence number at most two

Let $G$ be a graph, and let $\alpha(G)$ denote its independence number. A clique cover of $G$ is a collection of cliques whose union of covered edges contains every edge of $G$.

Erdős–Goodman–Pósa conjecture. If

$$\alpha(G)\le 2,$$

then $G$ has a clique cover of size at most $|G|$.

This is described as a long-standing conjecture, but the source does not identify its origin. Its status is therefore recorded as open.