  Using density matrix renormalization group simulations on open chains, we map
out the wave-vector in the incommensurate disordered phase of a realistic model
of Rydberg chains with $1/r^6$ interactions, and we locate and characterize the
points along the commensurate lines where the transition out of the period 3
and 4 phases is conformal. We confirm that it is 3-state Potts for the period-3
phase, and we show that it is Ashkin-Teller with $\nu\simeq 0.80$ for the
period-4 phase. We further show that close to these points the transition is
still continuous, but with a completely different scaling of the wave-vector,
in agreement with a chiral transition. Finally, we propose to use the conformal
points as benchmarks for Kibble-Zurek experiments, defining a roadmap towards a
conclusive identification of the chiral universality class.
