Andersen's rainbow path conjecture

Let $K_n$ be the complete graph on $n$ vertices, and let a proper edge-colouring be an edge-colouring in which edges of the same colour do not meet. A rainbow path is a path whose edges have pairwise distinct colours, and its length is its number of edges. Andersen's conjecture. Every proper edge-colouring of $K_n$ has a rainbow path with length $n-2$. The bound is best possible for powers of $2$, but the conjecture is not stated as resolved in the paper.