The harmonious labelling conjecture

Let $T$ be a tree with $n$ edges, and let $\mathbb{Z}_n$ denote the cyclic group of order $n$. A labelling is a map $\phi\colon V(T)\to \mathbb{Z}_n$; edge sums are the values $\phi(x)+\phi(y)$ for $xy\in E(T)$.

The harmonious labelling conjecture. For every $n$-edge tree $T$, there exists a labelling $\phi\colon V(T)\to \mathbb{Z}_n$ such that the edge sums $\phi(x)+\phi(y)$ are all distinct, and exactly one label is used on two vertices.

This is the cyclic additive labelling conjecture of Graham and Sloane. The paper states that it is confirmed for bounded-degree trees, but the unrestricted conjecture remains open.