NP-completeness conjecture for edge-clique cover on cographs

A cograph is a graph obtained from a single vertex by repeatedly applying disjoint union and complementation. For a graph $G$, the edge-clique cover problem asks whether the edges of $G$ can be covered by at most $k$ cliques, for a given integer $k$. Edge-clique cover conjecture. The edge-clique cover problem is NP-complete for cographs. This would establish NP-completeness for a graph class considered in the paper, complementing the polynomial-time results for the independence number of edge-clique graphs of cographs and distance-hereditary graphs. The supplied text gives no resolution of the conjecture.