Near-rainbow Hamilton cycle conjecture for properly coloured Dirac graphs

Let $G$ be a Dirac graph on $n$ vertices, meaning a graph with minimum degree at least $n/2$, and let its edge-colouring be proper if every pair of incident edges receives different colours. A Hamilton cycle is rainbow if all its edges receive distinct colours, and it has $r$ distinct colours when the set of colours on its edges has size $r$. Near-rainbow Hamilton cycle conjecture. Every proper edge-colouring of a Dirac graph on $n$ vertices contains a Hamilton cycle with at least $n/2-o(n)$ distinct colours. This asks for a linear lower bound without any global boundedness assumption; the supplied source gives no resolution.