Giant rainbow tree size conjecture near the phase transition

Let $G_c(n,p)$ be the random graph on $n$ vertices whose edges are independently present with probability $p$ and uniformly coloured with $c$ colours. Let $b1>0$, set $c=b1 n$, let $b5>0$ be sufficiently small, and set $p=\frac{1+b5}{n}$. Giant rainbow tree conjecture. With high probability, the largest rainbow tree in $G_c(n,p)$ has order

$$\left(2b5+O\left(b5^2\right)\right)n.$$

This would sharpen the proved estimate that the largest rainbow tree has order $\Theta(b5 n)$, matching the asymptotic order of the giant component in the corresponding ordinary random graph.