Rainbow path cover in properly edge-coloured graphs

Determine whether there exists a constant c > 0 such that the edges of every properly edge-coloured n-vertex graph can be covered by O(n) rainbow paths.

Background

Bonamy, Botler, Dross, Naia, and Skokan proved that every n-vertex graph has a separating path system of size at most 19n. Covering edges by O(n) rainbow paths in properly coloured graphs would imply an O(n) separating path system via Lovász’s path decomposition, unifying separation and colouring constraints.