The minimum conflict-free connection number conjecture for trees

Let $T$ be a tree of order $n$, and let $cfc(T)$ denote its conflict-free connection number, the smallest number of edge colors needed so that every two vertices are joined by a path containing a uniquely occurring color. The minimum conflict-free connection number conjecture for trees.

$$cfc(T)\geq \lceil \log_2 n\rceil.$$

Since $cfc(P_n)=\lceil \log_2 n\rceil$ is known, this conjecture says that the path has the least conflict-free connection number among all trees of order $n$.