Minimal-clique-cover conjecture for realizable graphs

Let $H$ be a realizable graph. A clique cover of $H$ is a collection of cliques whose union covers all edges of $H$, and let $n$ be the smallest number of cliques in such a cover. Minimal-clique-cover conjecture. There is always a digraph $G$ realizing $H$ with

$$|V_G|=n.$$

Equivalently, the theorem characterizing realizability continues to hold when the number of cliques is required to be minimal. This would make the decidability algorithm substantially more efficient by restricting the search to minimum-size clique covers; the source gives no resolution of the conjecture.