Stein's equitable perfect matching conjecture

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each part, and suppose its edges are partitioned into sets $E_1,\ldots,E_m$, where $m\leqslant n$. A perfect matching is a matching containing one edge incident with every vertex. Stein's equitable perfect matching conjecture. There exists a perfect matching $F$ in $K_{n,n}$ such that

$$|F\cap E_i|\geqslant\left\lfloor\frac{|E_i|}{n}\right\rfloor-1$$

for every $i$, with strict inequality holding for all but one value of $i$. This conjecture is implied by Stein's rainbow matching conjecture and is known in the case $m=3$ by the result cited in the source; the general case remains open.