On high discrepancy $1$-factorizations of complete graphs

We proved that for every sufficiently large $n$, the complete graph $K_{2n}$ with an arbitrary edge signing $\sigma: E(K_{2n}) \to \{-1, +1\}$ admits a high discrepancy $1$-factor decomposition. That is, there exists a universal constant $c > 0$ such that every edge-signed $K_{2n}$ has a perfect matching decomposition $\{\psi_1, \ldots, \psi_{2n-1}\}$, where for each perfect matching $\psi_i$, the discrepancy $\lvert \frac{1}{n} \sum_{e\in E(\psi_i)} \sigma(e) \rvert$ is at least $c$.