Bouchard–Hertz–Desaulniers deficiency bound for graphs

Let $G$ be a graph, let $V(G)$ denote its vertex set, and let $def(G)$ denote its deficiency.

Bouchard–Hertz–Desaulniers conjecture. For every graph $G$,

$$def(G)\leq \vert V(G)\vert.$$

The conjecture is still open, but it holds for regular graphs, some bipartite graphs, and outerplanar graphs. If true, it would imply upper bounds for the deficiency of triangle-free graphs and of graphs with at least three vertices.