The conjecture on nondegenerate induced cycles for sets with at least two primes

Let $S$ be a finite set of primes with $\#S\geq2$, and let $\mathcal{G}$ be its $S$-unit graph. A cycle is nondegenerate if its edge differences have no proper nonempty zero subsum, and induced if it has no edges between nonconsecutive vertices. The conjecture. There are nondegenerate induced cycles of $\mathcal{G}$ of every sufficiently large even length. This is presented as a belief after the corresponding assertion is refuted for singleton sets $S$; no resolution is supplied.