  The recent DESI 2024 Baryon Acoustic Oscillations (BAO) measurements combined
with the CMB data from the Planck 18 PR3 dataset and the Planck PR4+ACT DR6
lensing data, with a prior on the sum of the neutrino masses $\sum m_\nu>0$,
leads to a strong constraint, $\sum m_\nu<0.072$ eV, which would exclude the
inverted neutrino hierarchy and put some tension on even the standard
hierarchy. We show that actually this bound gets significantly relaxed when
combining the new DESI measurements with the HiLLiPoP+LoLLiPoP likelihoods,
based on the Planck 2020 PR4 dataset, and with supernovae datasets. We note
that the fact that neutrino masses are pushed towards zero, and even towards
negative values, is known to be correlated with the so-called $A_L$ tension, a
mismatch between lensing and power spectrum measurements in the Planck PR3
data, which is reduced by HiLLiPoP+LoLLiPoP to less than 1$\sigma$. We find
$\sum m_\nu<0.1$ eV and $\sum m_\nu<0.12$ eV, with the supernovae Pantheon+ and
DES-SN5YR datasets respectively. The shift caused by these datasets is more
compatible with the expectations from neutrino oscillation experiments, and
both the normal and inverted hierarchy scenarios remain now viable, even with
the $\sum m_\nu>0$ prior. Finally, we analyze neutrino mass bounds in an
extension of $\Lambda$CDM that addresses the $H_0$ tension, with extra fluid
Dark Radiation, finding that in such models bounds are further relaxed and the
posterior probability for $\sum m_\nu$ begins to exhibit a peak at positive
values.
