  Koperwas {\it et al.} showed in a recent paper, Phys. Rev. Lett. {\bf 111},
125701 (2013), that the dynamic susceptibility $\chi_4$ as estimated by
dielectric measurements for certain glass-forming liquids decreases
substantially with increasing pressure along a curve of constant relaxation
time. This observation is at odds with other measures of dynamics being
invariant and seems to pose a problem for theories of glass formation. We show
that this variation is in fact consistent with predictions for liquids with
hidden scale invariance: measures of dynamics at constant volume are invariant
along isochrones, called isomorphs in such liquids, but contributions to
fluctuations from long-wavelength fluctuations can vary. This is related to the
known non-invariance of the isothermal bulk modulus. Considering the version of
$\chi_4$ defined for the NVT ensemble, data from simulations of a binary
Lennard-Jones liquid show in fact a slight increase with increasing density.
This is a true departure from the formal invariance expected for this quantity.
