Exact minimum-degree conjecture for two-colored cycle partitions

Let $G$ be a graph on $n$ vertices whose edges are colored red or blue, and let $\delta(G)$ denote its minimum degree. A partition into cycles is a collection of vertex-disjoint monochromatic cycles spanning $V(G)$.

Exact cycle-partition conjecture. There exists an integer $n_0$ such that, whenever $n\geq n_0$ and

$$\delta(G)\geq\frac{3n-3}{4},$$

$G$ has a partition into a red cycle and a blue cycle.

The examples preceding this statement show that the threshold is sharp up to the strictness and rounding at the boundary. The source does not report a resolution of this exact asymptotic formulation.