Tournament path strengthening of the flash-and-rainbow conjecture

Let $l$ and $k$ be positive integers, and let $T$ be a tournament with $l^{k-1}+1$ vertices. A directed monochromatic path has all edges of one colour, while a directed rainbow path has pairwise distinct edge colours. Path strengthening. Every edge-colouring of $T$ contains a directed monochromatic path of length $l$ or a directed rainbow path of length $k$. This is a more restrictive variant of the tournament conjecture, which in the paper is formulated for directed walks; the source presents this path version as an open possibility because directed cycles create additional technical difficulties.