  The recent DESI Baryon Acoustic Oscillation measurements have led to tight
upper limits on the neutrino mass sum, potentially in tension with oscillation
constraints requiring $\sum m_{\nu} \gtrsim 0.06\,{\text{eV}}$. Under the
physically motivated assumption of positive $\sum m_{\nu}$, we study the extent
to which these limits are tightened by adding other available cosmological
probes, and robustly quantify the preference for the normal mass ordering over
the inverted one, as well as the tension between cosmological and terrestrial
data. Combining DESI data with Cosmic Microwave Background measurements and
several late-time background probes, the tightest $2\sigma$ limit we find
without including a local $H_0$ prior is $\sum m_{\nu}<0.05\,{\text{eV}}$. This
leads to a strong preference for the normal ordering, with Bayes factor
relative to the inverted one of $46.5$. Depending on the dataset combination
and tension metric adopted, we quantify the tension between cosmological and
terrestrial observations as ranging between $2.5\sigma$ and $5\sigma$. These
results are strenghtened when allowing for a time-varying dark energy component
with equation of state lying in the physically motivated non-phantom regime,
$w(z) \geq -1$, highlighting an interesting synergy between the nature of dark
energy and laboratory probes of the mass ordering. If these tensions persist
and cannot be attributed to systematics, either or both standard neutrino
(particle) physics or the underlying cosmological model will have to be
questioned.
