The multicolored cycle matching conjecture

Let $G$ be a cycle of length $2\ell$ whose edges are colored with $h$ colors. Let $M_0$ and $M_1$ be the perfect matchings consisting of the even and odd edges, respectively, and let $p_M\in\mathbb Z_+^h$ record the number of edges of each color in a matching $M$. Let $k\in\mathbb Z^h$ lie on the segment between $p_{M_0}$ and $p_{M_1}$. The multicolored cycle matching conjecture. There is a matching $M$ of size at least $\ell-h+2$ such that

$$p_M\leq k$$

component-wise and

$$\lVert p_M-k\rVert_1\leq h-2.$$

The paper presents this as a generalization of the corresponding three-color result and as a possible ingredient for extending the approximation guarantee to graphs with a constant number $h\geq4$ of colors. Its resolution is not given in the supplied text.