Li et al.'s maximum conflict-free vertex-connection number conjecture

Let $G$ be a connected graph of order $n$. The conflict-free vertex-connection number $vcfc(G)$ is the smallest number of colors required for a vertex-coloring in which every two vertices are joined by a path containing a color used exactly once. Li et al.'s conjecture.

$$vcfc(G)\leq vcfc(P_n).$$

The path $P_n$ is known to attain the upper bound among the graphs considered in the paper, and the conjecture asserts that it gives the maximum conflict-free vertex-connection number for every connected graph of order $n$.