Koljančić–Oblak Question 5.11 (arXiv:2604.21552v1, Section 5): nSSP witnesses for looped double paths

Question 5.11 of Koljančić and Oblak (arXiv:2604.21552v1, Section 5; journal p. 776) has an affirmative answer: every even double path with a nonempty loop set allows the non-symmetric Strong Spectral Property. We construct a symmetric one-loop witness using coprime branch continuants and prove injectivity of its characteristic-polynomial differential. Openness then supplies every prescribed nonempty loop set. The same argument gives a complete existence criterion in odd order: at least one loop must lie at an odd-numbered vertex. The witnesses can be chosen symmetric with positive integer nonzero entries. Carptopus (19 August 2026) previously proved the full existence criterion and its single-loop spectral characterization. We give an explicit integer construction, an arithmetic criterion for unit edge weights, and the three-way classification for the central-loop and interior-loop families.