The edge clique cover conjecture for graphs with independence number two

Let $G$ be a simple graph with $n$ vertices. A clique is a set of vertices inducing a complete graph, and the edge clique cover number $ecc(G)$ is the minimum number of cliques such that both endpoints of every edge lie in at least one of them. Write $balpha(G)$ for the independence number of $G$. Edge clique cover conjecture. If

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

then

$$ecc(G)\leq n.$$

This conjecture is a special case of the question whether every claw-free graph satisfies $ecc(G)\leq n$. Positive results were known for claw-free graphs with independence number greater than two, leaving the case $\alpha(G)=2$ as the conjectural case considered here.