Mohr–Pardey–Rautenbach conjecture on almost colour-balanced spanning forests

Let $K_n$ be the complete graph on $n$ vertices, and let $c:E(K_n)\rightarrow\{-1,+1\}$ be a balanced colouring, meaning that $c(K_n)=0$. Let $F$ be an $n$-vertex forest with maximum degree $\Delta$.

Mohr–Pardey–Rautenbach conjecture. There exists a copy $F'$ of $F$ in $K_n$ such that

$$\left|c(F')\right|\leq \tfrac{1}{2}(\Delta-1).$$

This conjecture asks how close to colour-balanced a spanning copy of an arbitrary forest can be in a balanced two-colouring of the complete graph. It generalises known results for perfect matchings, factors of paths, and spanning paths; its general case is presented as open in the source.