Conjecture on the temperature location of the intermediate clock-model regime

Let $\beta$ be the inverse temperature, let $\beta_c$ denote the critical inverse-temperature threshold, and let $q$ be the number of clock states. The notation $O(q^2)$ denotes the stated order-of-growth temperature scale for the discretized model. Intermediate-regime location conjecture. The intermediate regime conjectured in the continuum-of-massless-measures conjecture occurs once

$$\beta_c < \beta \leq O(q^2).$$

This is a more precise proposed location for the intermediate phase, following the discussion of Gibbsian and non-Gibbsian behavior for type-1 discretizations; the source gives no resolution.