Continuum of rotating states conjecture for the driven N-clock model

Let $\beta$ be sufficiently large, let $N(\beta)$ be a threshold depending on $\beta$, and consider the three-dimensional $N$-clock model with nonzero drift $d$.

Continuum rotating-states conjecture. For every sufficiently large $\beta$, there exists $N=N(\beta)$ such that, for every $N\geq N(\beta)$ and every $d>0$, the driven $N$-clock model has a continuum of distinct rotating states.

This conjecture extends the proposed rotating-state picture from a drift threshold to arbitrarily small positive drift when $N$ is sufficiently large. It is motivated by the expected similarity between the intermediate clock-model phase and the $XY$-model, and remains open.