Degree-bounded bipartite rainbow matching conjecture

Let $d>1$, and let $F_1,\ldots,F_k$ be bipartite graphs on a common vertex set. Write $\Delta(F_i)$ for the maximum vertex degree of $F_i$, and call a choice of pairwise vertex-disjoint edges $e_i\in F_i$ a rainbow matching.

Degree-bounded bipartite rainbow matching conjecture. If

$$\Delta(F_i)\leq d\quad\text{and}\quad |F_i|>(k-1)d$$

for every $i$, then the system $F_1,\ldots,F_k$ has a rainbow matching.

The source notes that the analogous assertion fails for $d=1$, while suggesting that the degree bound $d>1$ should suffice. No resolution is supplied in the source.