The two-piece monochromatic path and cycle covering conjecture for 3-colored complete graphs

Let $K_n$ be a complete graph whose edges are colored with three colors. A monochromatic path or cycle is a path or cycle all of whose edges have one color, and the two pieces are required to be vertex disjoint. The two-piece covering conjecture. There are two vertex-disjoint monochromatic paths, or two vertex-disjoint monochromatic cycles, covering at least

$$\frac{6n}{7}$$

of the vertices. The paper proves only a weaker matching statement, so this remains open there.