  We reassess the viability of a cosmological model including a fourth
additional sterile neutrino species that self-interacts through a new
pseudoscalar degree of freedom. We perform a series of extensive analyses
fitting various combinations of cosmic microwave background (CMB) data from
Planck, the Atacama Cosmology Telescope (ACT) and the South Pole Telescope
(SPT), both alone and in combination with Baryon Acoustic Oscillation (BAO) and
Supernova Ia (SnIa) observations. We show that the scenario under study,
although capable to resolve the Hubble tension without worsening the so-called
$S_8$ tension about the growth of cosmic structures, is severely constrained by
high-multipole polarization data from both Planck and SPT. Intriguingly, when
trading Planck TE-EE data for those from ACT, we find a $\gtrsim 3 \sigma$
preference for a non-zero sterile neutrino mass, $m_s=3.6^{+1.1}_{-0.6}$ eV (68
% C.L.), compatible with the range suggested by longstanding short-baseline
(SBL) anomalies in neutrino oscillation experiments. The pseudoscalar model
provides indeed a better fit to ACT data compared to $\Lambda$CDM
($\Delta\chi^2 \simeq -5$, $\Delta \rm{AIC}=-1.3$), although in a combined
analysis with Planck the $\Lambda$CDM model is still favoured, as the
preference for a non-zero sterile neutrino mass is mostly driven by ACT
favouring a higher value for the primordial spectral index $n_s$ with respect
to Planck. We show that the mild tension between Planck and ACT is due to the
different pattern in the TE and EE power spectra on multipoles between $350
\lesssim \ell \lesssim 1000$. We also check the impact of marginalizing over
the gravitational lensing information in Planck data, showing that the model
does not solve the CMB lensing anomaly. Future work including higher precision
data from current and upcoming CMB ground-based experiments will be crucial to
test these results.
