Conjecture on a continuum of massless Gibbs measures for high-dimensional clock models

Consider the nearest-neighbor $q$-state clock model in spatial dimension $d\geq 3$, at inverse temperature below the critical-temperature threshold and with $q$ sufficiently large. A massless Gibbs measure is a Gibbs measure exhibiting massless, rather than massive, long-range behavior. Continuum-of-massless-measures conjecture. There exists a continuum of massless Gibbs measures in an intermediate regime below the critical temperature for large-$q$ clock models in three and more dimensions. The conjecture asks whether the intermediate phase suggested by analogous results for continuous-spin models has continuously many Gibbs states; the source says that numerical evidence is inconclusive and that even the existence of the intermediate phase is under doubt.