  The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the
minimum number of colors needed to properly color the edges of $G$ such that no
path or cycle of length four is bi-colored. A multigraph $G$ is star
$k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal
[Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every
subcubic multigraph is star $7$-edge-colorable, and conjectured that every
subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and
Raspaud considered the list version of this problem for simple graphs and
proved that every subcubic graph with maximum average degree less than $7/3$ is
star list-$5$-edge-colorable. It is known that a graph with maximum average
degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we
prove that every subcubic multigraph with maximum average degree less than
$12/5$ is star $5$-edge-colorable.
