Rainbow Caccetta–Häggkvist conjecture for edge-set families

Let $G$ be an undirected graph, and let $F_1,F_2,\ldots,F_n$ be sets of edges in $G$. For a family of edge sets, let $rg(F_1,\ldots,F_n)$ denote the length of its shortest rainbow cycle. Rainbow Caccetta–Häggkvist conjecture. If each $F_i$ has size at most $k$, then

$$rg(F_1,F_2,\ldots,F_n)\leq \big\lceil \frac{n}{k}\big\rceil.$$

This is a rainbow analogue of the Caccetta–Häggkvist conjecture, relating prescribed edge colours to short rainbow cycles. The paper presents it as a conjecture suggested by the first author; the stated general form remains open.