Rainbow directed cycle-or-path conjecture for proper arc-colourings

Let $\overleftrightarrow{K_n}$ be the digraph obtained from the complete graph $K_n$ by replacing each edge with two oppositely directed arcs and adding a directed loop at each vertex. A proper arc-colouring is a colouring in which no two arcs of the same colour have a common head or a common tail. Rainbow directed cycle-or-path conjecture. Every proper arc-colouring of $\overleftrightarrow{K_n}$ admits a rainbow directed cycle or path of length at least $n-1$. This conjecture is motivated by bounds for rainbow directed cycles and by the study of connected rainbow subgraphs with maximum in-degree and out-degree at most one; it remains open.