Aharoni–Howard rainbow matching conjecture

Let $H_1,\dots,H_k$ be $r$-uniform hypergraphs on the same set of $n$ vertices. Define $f(n,r,k)$ to be the smallest $m$ such that every $r$-uniform hypergraph on $n$ vertices with $m$ edges has a matching of size $k$.

Aharoni–Howard's conjecture. If each $H_i$ has $f(n,r,k)$ hyperedges, then there is a rainbow matching

$$\{e_1,\dots,e_k\},\qquad e_i\in E(H_i)\text{ for }i=1,\dots,k.$$

This is a rainbow version of the Erdős–Ko–Rado matching threshold. It is proved for balanced $r$-partite hypergraphs when $r=3$, while the general statement remains open.