Tight cycle-partition conjecture for dense edge-coloured graphs

For $r\geq2$ and $\delta\in(0,1/2)$, let

$$cp_r(\delta):=\limsup_{n\to\infty}\max_{G\in\mathcal{G}(n,r,\delta)}cp(G),$$

where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $cp(G)$ is the smallest number of vertex-disjoint monochromatic cycles partitioning $V(G)$. Tight cycle-partition conjecture. There exists $K>0$ such that, for all $r\geq2$ and $\delta\in(0,1/2)$,

$$cp_r(\delta)\leq Kr\left\lceil\frac{r}{\log(1/\delta)}\right\rceil.$$

The conjecture would improve the paper's $\mathcal{O}(r\log r\lceil r/\log(1/\delta)\rceil)$ upper bound to the matching order suggested by the lower bound. It is known for $\delta=\Omega(1)$, but remains open in general.