Conjecture on rainbow perfect matching packings in random bipartite graphs

Let $n$ be a positive integer, let $p$ satisfy

$$p=\omega\left(\frac{\log n}{n}\right),$$

and let $\mathcal{G}_c(K_{n,n},p)$ denote the random graph on the complete bipartite graph $K_{n,n}$ whose present edges receive a random edge-coloring with $c$ colors. A matching is rainbow if all its edges have distinct colors. Rainbow perfect matching packing conjecture. There exists $c=(1+o(1))n$ such that, with high probability, a graph $H\sim\mathcal{G}_c(K_{n,n},p)$ contains $(1-o(1))np$ edge-disjoint rainbow perfect matchings. This conjecture proposes asymptotically optimal parameters for packing rainbow perfect matchings, improving the proved bound with $c=3n$ and $(1-o(1))np/6$ matchings; its status is not resolved in the supplied source.