Brualdi–Quinn Massey's strong chromatic index conjecture for bipartite graphs

Let $G$ be an $(a,b)$-bipartite graph, meaning that the two vertex classes of $G$ have maximum degrees at most $a$ and $b$, respectively. Brualdi–Quinn Massey's conjecture. The strong chromatic index satisfies

$$\chi'_s(G)\leq ab.$$

This conjecture refines the earlier conjecture of Faudree, Gyárfás, Schelp, and Tuza that every bipartite graph $G$ satisfies $\chi'_s(G)\leq \Delta(G)^2$. Its status is not specified in the supplied text.