Chromatic index, treewidth and maximum degree

We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $\mathrm{\Delta }(G)\ge k+\sqrt{k}$ satisfies ${\chi }^{\mathrm{\prime }}(G)=\mathrm{\Delta }(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\ge 4$ and maximum degree $2k-1$ satisfies ${\chi }^{\mathrm{\prime }}(G)=\mathrm{\Delta }(G)$, improving an old result of Vizing.