The square-root discrepancy conjecture for representative perfect matchings in bipartite graphs

Let $K_{n,n}$ be the complete bipartite graph with two parts of size $n$, and let $h\colon E(K_{n,n})\to\mathbb{R}^k$ satisfy

$$\|h(e)\|_1\leq 1$$

for every edge $e\in E(K_{n,n})$. For a perfect matching $M$, let $f_h(M)$ denote the representative-matching deviation associated with $h$.

Square-root discrepancy conjecture. There is a perfect matching $M$ of $K_{n,n}$ satisfying

$$f_h(M)=O(\sqrt{k}).$$

The conjecture seeks to improve the $O(k^2)$ upper bound for representative matchings in bipartite graphs toward the known lower bound of $\sqrt{k/2}$. It remains open in the supplied source.