Smyth's conjecture on linear relations among Galois conjugates

Let $n$ be a positive integer and let $(a_1,\dots,a_n)\in\mathbb{Q}^n$ satisfy

$$|a_i|\leq\sum_{j\neq i}|a_j|\quad\text{for all }i,$$

and suppose every prime $p$ divides at most $n-2$ of the $a_i$.

Smyth's conjecture. There exist Galois conjugates $\gamma_1,\dots,\gamma_n$ such that

$$\sum_{i=1}^n a_i\gamma_i=0.$$

This conjecture asserts that the two necessary conditions identified by Smyth are also sufficient over $\mathbb{Q}$. The paper proves the analogous statement over $\mathbb{F}_q(t)$, while the number-field case remains the subject of the conjecture.