Javadi–Maleki–Omoomi conjecture on local clique cover numbers

Let $G$ be a graph on $n$ vertices, and let $\operatorname{lcc}(G)$ denote its local clique cover number. Let $\overline{G}$ be the complement of $G$. Javadi–Maleki–Omoomi conjecture. For every graph $G$ on $n$ vertices,

$$\operatorname{lcc}(G)+\operatorname{lcc}(\overline{G})\leq n.$$

This is a Nordhaus–Gaddum-type inequality for the local clique cover number. It was proposed by R. Javadi, Z. Maleki and B. Omoomi in 2012; the supplied source gives no resolution, so its status is open.