The decomposition conjecture for class II graphs

Let $G$ be a graph with maximum degree $\Delta$ and edge-chromatic number $\chi'(G)=\Delta+k$, and let $p,q$ be positive integers satisfying

$$p+q=\Delta+k$$

with $p,q\leq\Delta$. A graph is class I if its edge-chromatic number equals its maximum degree. Decomposition conjecture. The graph $G$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that

$$\Delta(H_1)=p,\qquad \Delta(H_2)=q.$$

This would generalize the preceding theorem, which gives a decomposition with maximum degrees $\Delta$ and $k$; the proposed statement allows any two positive maximum degrees at most $\Delta$ whose sum is $\Delta+k$.