  A strong edge coloring of a graph $G$ is a proper edge coloring in which each
color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey
proposed a conjecture that every bipartite graph without $4$-cycles and with
the maximum degrees of the two partite sets $2$ and $\Delta$ admits a strong
edge coloring with at most $\Delta+2$ colors. We prove that this conjecture
holds for such graphs with $\Delta=3$. We also confirm the conjecture proposed
by Faudree et al. for subcubic bipartite graphs.
