The path extremal conjecture for conflict-free vertex-connection number

Let $G$ be a connected graph of order $n$. The path extremal conjecture.

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

Here $vcfc(G)$ denotes the conflict-free vertex-connection number of $G$, and $P_n$ is the path on $n$ vertices. The conjecture proposes that among connected graphs of a fixed order, the path has maximum conflict-free vertex-connection number. The surrounding results establish upper bounds for trees and connected graphs in terms of radius and order, but do not settle this extremal assertion.