Multicolour Leader–Long antipodal path conjecture

Let $Q_n$ be the $n$-dimensional hypercube. A geodesic path is a shortest path between its endpoints, and a colour change occurs when consecutive edges receive different colours. Multicolour Leader–Long conjecture. For every $k\in\mathbb{N}$, in every $(k+1)$-colouring of the edges of $Q_n$, there exists a geodesic path between antipodal vertices with at most $k$ colour changes. The source presents this as the analogue of the two-colour conjecture.