The cyclotomic-integer conductor-length conjecture

Let $\alpha$ be a cyclotomic integer, meaning an algebraic integer expressible as a sum of roots of unity. Let $l(\alpha)$ be the smallest number of roots of unity occurring in any such representation, and let $c(\alpha)$ be the smallest positive integer such that $\mathbb{Q}(\alpha)\subseteq\mathbb{Q}_{c(\alpha)}$, where $\mathbb{Q}_n$ is the field generated by a primitive $n$th root of unity. The conductor-length conjecture. One has

$$[\mathbb{Q}_{c(\alpha)}:\mathbb{Q}(\alpha)] \leq l(\alpha).$$

This is the purely number-theoretic reformulation of the element-level question of Hung and Tiep. The paper's abstract states that the question is answered negatively and that all counterexamples with length at most $4$ are characterized; it also studies the growth of the degree as a function of the length. Thus this conjecture is refuted.