Three-colour random bipartite tree-cover threshold conjecture

Let $G\sim G(n,n,p)$ be a random bipartite graph with two parts of size $n$, and let $\operatorname{tc}_3(G)$ be the minimum number of monochromatic trees whose vertices cover $G$ in every 3-edge-colouring. Three-colour random bipartite conjecture. There exists a constant $C$ such that if

$$p\ge C(\log n/n)^{1/4},$$

then with high probability

$$\operatorname{tc}_3(G)\le 5.$$

This is proposed as the three-colour analogue of the two-colour threshold question and remains open.