Pardey–Rautenbach conjecture on almost colour-balanced perfect matchings

Let $n$ and $k$ be positive integers, let $K_{2kn}$ be the complete graph on $2kn$ vertices, and let $[k]=\{1,2,\dotsc,k\}$. A colouring $c\colon E(K_{2kn})\to [k]$ is colour-balanced when every colour is used on the same number of edges. For a perfect matching $M$ of $K_{2kn}$, define

$$f(M)\coloneqq \sum_{i=1}^k\left|\left|c^{-1}(i)\cap M\right|-n\right|.$$

Pardey–Rautenbach conjecture. If $c\colon E(K_{2kn})\to [k]$ is colour-balanced, then there is a perfect matching $M$ of $K_{2kn}$ such that

$$f(M)\leq \mathcal{O}(k^2).$$

The conjecture seeks a bound depending only on the number of colours, rather than on the number of vertices. Pardey and Rautenbach proposed it after showing that an exactly colour-balanced perfect matching need not exist for three colours; the paper improves the previous bound, which depended on $n$, to an explicit bound of $4^{k^2}$, but does not establish the conjectured quadratic dependence.