The bounded-discrepancy conjecture for spanning forests in zero-sum complete graphs

Let $K_n$ be a complete graph, let $c:E(K_n)\to\{-1,1\}$ be a zero-sum labeling satisfying

$$c(E(K_n))=\sum_{e\in E(K_n)}c(e)=0,$$

and let $F$ be a spanning forest of $K_n$. Write $\Delta(F)$ for the maximum degree of $F$. The bounded-discrepancy conjecture. There is an isomorphic copy $F'$ of $F$ in $K_n$ such that

$$|c(E(F'))|\leq\frac{\Delta(F)-1}{2}.$$

The conjecture would strengthen the general bound $|c(E(F'))|\leq\Delta(F)+1$ and is verified in the paper for stars; the theorem proved there also gives the claimed bound for spanning forests with sufficiently large maximum degree. The general case remains open.