It would seem that the ions are probably emitting photons at frequencies distributed about the reference frequency [tex]\eta_o[/tex] in the moving ion frame --- all photons are probably not exactly at [tex]\eta_o[/tex] due to, e.g., Stark and/or Zeeman effects.
However, the measurements --- to detect when the lasers excite the resonance --- is done in the laboratory frame at frequency [tex]\eta_o[/tex] (use of narrow prefilters centered at [tex]\eta_o[/tex], e.g., as described in detail in Ref [14] of the article).
Therefore, the experiments appear to measure the lamb dip in specific set of photons at [tex]\eta_o[/tex] with respect to the laboratory frame, which could bias the results... since these would not correspond to [tex]\eta_o[/tex] with respect to the moving ion frame due to Doppler shifts.