Colourful almost-spanning tree embedding conjecture for Dirac graphs

Let $c$ satisfy $1/2<c\leqslant1$, and let $G$ be an $n$-vertex graph with minimum degree $\delta(G)\geqslant cn$. Colourful tree embedding conjecture. There exists a constant $d>0$ such that, for every tree $T$ on $n$ vertices with maximum degree $\Delta(T)\leqslant dn/\log n$, every proper edge-colouring of $G$ contains a copy of $T$ using at least $cn-o(n)$ distinct colours. The conjecture is a colourful analogue of bounded-degree tree-embedding results for Dirac graphs; the supplied text gives no resolution evidence, so it remains open.