Aharoni–Howard conjecture on rainbow matchings

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$, and let $\binom{[n]}{k}$ be the family of $k$-subsets of $[n]$. Let $\mathcal F=\{F_1,\ldots,F_t\}$ be a family of subsets of $\binom{[n]}{k}$, and call a set of pairwise disjoint edges, one from each $F_i$, a rainbow matching. Aharoni–Howard conjecture. If

$$e(F_i)>\max\left\{\binom{n}{k}-\binom{n-t+1}{k},\binom{kt-1}{k}\right\}$$

for every $1\le i\le t$, then $\mathcal F$ admits a rainbow matching. This is the family version of the Erdős matching conjecture and seeks a sharp edge-density condition guaranteeing a rainbow matching; the source provides no evidence of resolution.