Rainbow cycles for families of matchings

Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous Caccetta-H\"aggkvist conjecture, proposed by the first author, is that if in an coloring of the edge set of an $n$-vertex graph by $n$ colors, in which each color class is of size $k$, the rainbow girth is at most $\lceil \frac{n}{k} \rceil$. In the known examples for sharpness of this conjecture the color classes are stars, suggesting that when the color classes are matchings, the result may be improved. We show that the rainbow girth of $n$ matchings of size at least 2 is $O(\log n)$.