The clique-cover chromatic-index upper-bound conjecture for random graphs

Let $G$ be sampled from the binomial random graph model ${\mathcal G}(n,p)$, where $p$ is a constant with $0<p<1$. Let $\theta'_0(G)$ be the minimum, over all clique covers of $G$, of the chromatic index of the clique cover; equivalently, it is the smallest number of subgraphs with complete-graph components whose union covers $E(G)$. Clique-cover chromatic-index conjecture. There exists a constant $c_4=c_4(p)>0$ such that, with high probability as $n\to\infty$,

$$\theta'_0(G)<c_4\frac{n}{\log n}.$$

The conjecture proposes an $O(n/\log n)$ upper bound for the clique-cover chromatic index of a typical dense random graph. The source motivates it by suggesting that the Frieze--Reed method for random graph thickness may yield the correct order, but gives no proof.