High sensitivity to mass-ratio variation in deep molecular potentials

Molecular vibrational transitions are prime candidates for model-independent searches for variation of the proton-to-electron mass ratio. Searches for present-day variation achieve highest sensitivity with deep molecular potentials. We identify several high-sensitivity transitions in the deeply bound ${\rm O}_2^+$ molecular ion. These transitions are electric-dipole forbidden and thus have narrow linewidths. The most sensitive transitions take advantage of an accidental degeneracy between vibrational states in different electronic potentials. We suggest experimentally feasible routes to a measurement with uncertainty exceeding current limits on present-day variation in $m_p/m_e$.