Vizing's average degree conjecture for edge-Δ-critical simple graphs

Let $G$ be a finite, undirected, loopless simple graph, let $|G|$ denote its number of vertices, let $\Delta=\Delta(G)$ be its maximum degree, and let $\overline{d}(G)$ be its average degree. The graph $G$ is $\Delta$-critical if it is critical and has chromatic index $\chi'(G)=\Delta+1$. Vizing's average degree conjecture. Every $\Delta$-critical simple graph $G$ satisfies

$$\overline{d}(G) \ge \Delta-1+\frac{3}{|G|}.$$

The conjecture gives a lower bound on the density of edge-chromatic critical simple graphs. A number of results improve lower bounds, but the conjectured bound remains open.