The rainbow 2-factor conjecture for properly colored complete graphs

Let $K_{2n}$ be a complete graph with a proper edge-coloring using exactly $2n-1$ colors. A multicolored $2$-factor is a $2$-factor whose edges have pairwise distinct colors. The rainbow 2-factor conjecture. Every proper edge-coloring of $K_{2n}$ by $2n-1$ colors contains a multicolored $2$-factor on either $2n-1$ or $2n$ vertices.

This is an anti-Ramsey-type strengthening of the fact that sufficiently many colors force a multicolored $1$-factor. The two allowed orders account for parity issues, and the paper proposes the assertion as an open problem.