Decomposing multigraphs into two Class I subgraphs with prescribed maximum degrees

Let $G$ be a finite loopless multigraph with maximum degree $Δ$ and $χ'(G)=Δ+k$, where $k\ge0$. We prove that for every pair of positive integers $p,q$ satisfying $$p+q=Δ+k\qquad\text{and}\qquad p,q\leΔ,$$ the graph $G$ admits an edge decomposition into two spanning Class I subgraphs $H_p,H_q$ with