Jakobsen's bounded-order conjecture for critical multigraphs

Let $G$ be a critical multigraph, meaning that $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$, and let $m$ be an odd integer with $m\geq 3$.

Jakobsen's bounded-order conjecture. If

$$\chi'(G)>\frac{m\Delta(G)+(m-3)}{m-1},$$

then $G$ has at most $m-2$ vertices.

The source attributes this conjecture to Jakobsen and states that Andersen proved it weaker than the Goldberg–Seymour conjecture. It is therefore solved by the result established in the paper.