Even factors in edge-chromatic-critical graphs with a small number of divalent vertices

A finite simple connected graph $G$ with maximum degree $k$ is $k$-critical if it has chromatic index $\chi'(G)=k+1$ and $\chi'(G-e)=k$ for every edge $e\in E(G)$. Bej and the first author raised the question whether every $k$-critical graph has an even factor. We prove that every $k$-critical graph with at most $2k-6$ vertices of degree 2 has an even factor.