Mohr–Pardey–Rautenbach's zero-sum tree-factor conjecture

Let $K_n$ be the complete graph on $n$ vertices, and let $T$ be a tree of order $k$. A labeling $\mathbf{c}:E(K_n)\to\{-1,1\}$ is 0-sum if the sum of the labels of all edges of $K_n$ is zero; a spanning forest whose components are all isomorphic to $T$ is a $T$-factor, and it is 0-sum if the sum of the labels of its edges is zero.

Mohr–Pardey–Rautenbach's conjecture. Fix integers $k,n\geq 1$ such that $\binom{n}{2}$ and $(k-1)n/k$ are both even integers. If $\mathbf{c}:E(K_n)\to\{-1,1\}$ is a 0-sum labeling and $n$ is sufficiently large in terms of $k$, then $K_n$ has a 0-sum $T$-factor.

The conjecture concerns zero-discrepancy decompositions of a balanced complete graph into copies of a fixed tree. The supplied text says that the conjecture is refuted by the preceding construction, so it is not an open conjecture.