The conjecture that three-dimensional ε-isospectral lens spaces are ε-spin-isometric

Let two three-dimensional lens spaces be ε-isospectral if their Dirac spectra agree up to the equivalence encoded by ε, and let them be ε-spin-isometric if there is an ε-compatible spin isometry between them. The ε-isospectrality conjecture. Any two three-dimensional ε-isospectral lens spaces are ε-spin-isometric. The conjecture proposes that, in dimension three, the Dirac spectrum determines a lens space up to the corresponding spin-isometry. It is motivated by computer calculations for parameters up to $q\leq 10^4$, but the general claim remains open.