The constant-deviation conjecture for colour-balanced perfect matchings

Let $K_{2kt}$ be the complete graph on $2kt$ vertices, and let $c\colon E(K_{2kt})\to [k]$ be a $k$-edge-colouring. Call $c$ colour-balanced if every colour appears equally often, and for a perfect matching $M$ define

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

Constant-deviation conjecture. There exists an absolute constant $C$ such that for all positive integers $k$ and $t$, every colour-balanced $k$-edge-colouring of $K_{2kt}$ admits a perfect matching $M$ satisfying $f_c(M)<C$.

This conjecture proposes that the colour imbalance can be bounded independently of both the number of colours and the matching size. The source presents it as an open problem, contrasting it with an $O(k^2)$ upper bound and a constant lower bound for matchings in complete graphs.