Colourful Hamilton cycle packing conjecture for Dirac graphs

Let $c$ satisfy $1/2<c\leqslant1$, let $G$ be an $n$-vertex graph with minimum degree $\delta(G)\geqslant cn$, and suppose the edges of $G$ are properly edge-coloured. Colourful Hamilton cycle packing conjecture. The graph $G$ contains at least $n/8$ Hamilton cycles, each using at least $cn-o(n)$ distinct colours. The conjecture is motivated by results on edge-disjoint Hamilton cycle packings in uncoloured graphs; the supplied text gives no resolution evidence, so it remains open.