Given a graph G = (V, E) whose vertices have been properly coloured, we say that a path in G is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy Theorem that every properly coloured graph contains a colourful path on \chi(G) vertices, where \chi(G) is the chromatic number of G. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on \chi(G) vertices.