Ruzsa's conjecture on nondegenerate induced cycles in S-unit graphs

Let $S$ be a finite nonempty set of primes. Let $\mathcal{G}$ be the graph with vertex set $\mathbb{Q}$ in which two rationals are adjacent when their difference is an $S$-unit. A cycle of $\mathcal{G}$ is nondegenerate if, writing its successive edge differences as $u_1,\ldots,u_n$, no proper nonempty subsum of $u_1+\cdots+u_n=0$ vanishes. It is induced if no nonconsecutive vertices are adjacent. Ruzsa's conjecture. There are nondegenerate induced cycles of $\mathcal{G}$ of every sufficiently large even length. The conjecture is refuted: when $S=\{p\}$, such cycles exist precisely in lengths $n$ satisfying $n\equiv 2\pmod{p-1}$, so not in every sufficiently large even length.