Pósa-type degree-sequence conjecture for monochromatic cycle partitions

Let $d_1\leq\dots\leq d_n$ be the degree sequence of a graph $G$. The degree-sequence conjecture. There is a function $f$ such that, for every $\beta>0$, every integer $r\geq 2$, and all sufficiently large $n$, if

$$d_i\geq i+\beta n$$

for every $i\leq n/2$, then every $r$-edge-colouring of $G$ admits a partition of $V(G)$ into at most $f(r)$ monochromatic cycles. This conjecture extends Pósa-type Hamiltonicity conditions to coloured cycle partitions; an approximate solution is known for $r=2$, but the general assertion remains open.