The complexity trichotomy for circular-colouring mixing

Let $G_{p,q}$ denote the graph used as the target for $(p,q)$-colourings, and let the textsc{G_{p,q}}-Mixing problem ask whether the reconfiguration graph of $(p,q)$-colourings of an input graph is connected. Circular-colouring mixing complexity conjecture. If

$$\frac{p}{q}=2,$$

then \textsc{G_{p,q}}-Mixing is in P; if

$$2<\frac{p}{q}<4,$$

then it is co-NP-complete; and if

$$\frac{p}{q}\geq 4,$$

then it is PSPACE-complete. The first two cases extend known complexity results for odd-cycle mixing and planar inputs, while the full classification remains open, particularly the PSPACE-completeness claim for ratios at least $4$.