  The Oef category gathers rapidly rotating and evolved O-stars displaying a
centrally reversed He II 4686 emission line. The origin of the variability of
their photospheric and wind spectral lines is debated, with rotational
modulation or pulsations as the main contenders. To shed new light on this
question, we analysed high-quality and high-cadence TESS photometric time
series for five Oef stars. We also collected a new time series of spectra for
one target (lambda Cep) which had been the subject of specific debates in the
last years. These observations reveal the variety of Oef behaviours. While
space-based photometric data reveal substantial red noise components in all
targets, only zeta Pup seems to display a long-lived periodicity. In our
sample, stars exhibit a dominant signal at low frequencies but it appears
relatively short-lived. This is reminiscent of rotational modulations by
transient photospheric spots, though this scenario is challenged by the case of
HD 14442, whose 1.230 d$^{-1}$ signal significantly exceeds the critical
rotational frequency. In parallel, no evidence of persistent p-mode non-radial
pulsations is found in either photometry or spectroscopy of the stars, only
temporary excitation of g-mode pulsations could offer an alternative
explanation for the dominant signals. Finally, the revised luminosities of the
stars using GAIA-DR2 show that they are not all supergiants as zeta Pup. The
question then arises whether the Oef peculiarity denotes a homogeneous class of
objects after all.
