Akbari–Etesami–Mahini–Mahmoody conjecture on rainbow Hamilton cycles

Let $K_n$ be the complete graph on $n$ vertices, and let $\chi'(K_n)$ denote its edge-chromatic number. A properly edge-coloured graph is one in which adjacent edges receive distinct colours. Akbari–Etesami–Mahini–Mahmoody's conjecture. Every properly edge-coloured $K_n$ with $\chi'(K_n)$ colours has a Hamilton cycle using $O(\log n)$ colours.

This conjecture asks for a Hamilton cycle using very few colours in an optimally properly edge-coloured complete graph. The source presents it as a related open problem.