The path extremal conjecture for the game cordiality number of trees

Let $T$ be a tree of order $n$, and let $P_n$ denote the path on $n$ vertices. Path extremal conjecture. The game cordiality number satisfies

$$c_g(T)\le c_g(P_n).$$

This conjecture proposes that paths are the worst-case trees for the cordiality game, in contrast to the upper bound proved earlier in the paper. Its status is not specified in the source.