Pardey and Rautenbach's bounded-deviation conjecture for colour-balanced 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

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

Pardey and Rautenbach's conjecture. For all integers $k\geq 2$ and $t\geq 1$, every colour-balanced $k$-edge-coloured $K_{2kt}$ admits a perfect matching $M$ satisfying $f_c(M)=O(k^2)$.

This asks for a uniformly controlled colour imbalance in a perfect matching, extending the known bound $f_c(M)\leq 2$ when $k=3$. The conjecture remains open in the supplied source.