Average degree conjecture for critical multigraphs

Let $G$ be a finite, undirected, loopless multigraph. Write $\Delta=\Delta(G)$ for its maximum degree, $\chi'(G)$ for its chromatic index, and $\overline{d}(G)$ for its average degree. The graph $G$ is $\Delta$-critical when it is critical and $\chi'(G)=\Delta+1$, where critical means that every proper subgraph $H\subset G$ has $\chi'(H)<\chi'(G)$. The average degree conjecture. Every $\Delta$-critical graph $G$ with $\Delta\ge2$ satisfies

$$\overline{d}(G)\ge\frac{2\Delta+2}{3}.$$

The paper proves this bound for $\Delta\in\{2,3,4,5,6,7,8\}$, while the conjecture remains open in general; the proposed bound is stated to be best possible.