The clique cover bound for triad-free graphs

Let $G$ be a graph on $n$ vertices, and let $\alpha(G)$ denote its independence number, the maximum size of a set of pairwise nonadjacent vertices. The graph is triad-free when $\alpha(G)\leq 2$. Clique cover conjecture for triad-free graphs. If $\alpha(G)\leq 2$, then

$$\operatorname{cc}(G)\leq n.$$

Here $\operatorname{cc}(G)$ is the clique cover number of $G$, the minimum number of cliques whose edges cover all edges of $G$. The paper notes that this would improve the known asymptotic upper bound for triad-free graphs; the claim remains open.