  Gamma~Cas stars are early-type Be stars that exhibit an unusually hard and
bright thermal X-ray emission. One of the proposed scenarios to explain these
properties postulates the existence of a neutron star companion in the
propeller stage, during which the magnetosphere of a rapidly rotating neutron
star repels infalling material. To test this model, we examined the fluorescent
Fe K$\alpha$ emission line at $\sim 6.4$\,keV in the X-ray spectra of
$\gamma$~Cas stars, which offers a powerful diagnostic of both the primary
source of hard X-rays and the reprocessing material. We computed synthetic line
profiles of the fluorescent Fe K$\alpha$ emission line in the framework of the
propelling neutron star scenario. Two reservoirs of material contribute to the
fluorescence in this case: the Be circumstellar decretion disk and a shell of
cool material that surrounds the shell of X-ray-emitting plasma around the
putative propelling neutron star. We analysed the synthetic line profiles and
expected equivalent widths of the lines for three well-studied $\gamma$~Cas
stars. The predicted line strengths fall short of the observed values by at
least an order of magnitude. Pushing the model parameters to reproduce the
observed line strengths led to column densities towards the primary X-ray
source that exceed the observationally determined values by typically a factor
of 20, and would further imply a higher X-ray luminosity than observed. The
strengths of the observed Fe K$\alpha$ fluorescent emission lines in
$\gamma$~Cas stars are inconsistent with the expected properties of a propeller
scenario as proposed in the literature.
