Hajebi's conjecture on subquadratic clique covers of K_{t,t}-free graphs

Let $t\ge 1$ be an integer. A graph is $K_{t,t}$-free if it has no induced copy of the complete bipartite graph $K_{t,t}$, and a clique cover is a collection of cliques covering every edge.

Hajebi's conjecture. For every integer $t\ge 1$, there exists $\varepsilon>0$ such that every $K_{t,t}$-free graph $G$ has a clique cover of size

$$O\left(|G|^{2-\varepsilon}\right).$$

The conjecture is attributed to Sepehr Hajebi, who proposed it at a conference in Lyon. The paper proves a related bound with $\varepsilon=1/(2t)$, so the proposed statement is solved by the result described in the source.