Goldberg's conjecture on total chromatic numbers

Let $G$ be a graph, let $\Delta(G)$ be its maximum degree, let $\chi'(G)$ be its chromatic index, and let $\chi”(G)$ be its total chromatic number. A graph is edge-chromatic critical if $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$.

Goldberg's conjecture. If

$$\chi'(G)\geq\Delta(G)+3,$$

then

$$\chi”(G)=\chi'(G).$$

Goldberg observed that the equality holds for edge-chromatic critical graphs under the Goldberg–Seymour conjecture, while the paper explicitly states that the conjecture does not hold for all edge-chromatic critical graphs. Thus this conjecture is refuted.