The sensitivity of cosmology to the total neutrino mass scale $Σm_ν$ is approaching the minimal values required by oscillation data. We study quantitatively possible tensions between current and forecasted cosmological and terrestrial neutrino mass limits by applying suitable statistical tests such as Bayesian suspiciousness, parameter goodness-of-fit tests, or a parameter difference test. In particular, the tension will depend on whether the normal or the inverted neutrino mass ordering is assumed. We argue, that it makes sense to reject inverted ordering from the cosmology/oscillation comparison only if data are consistent with normal ordering. Our results indicate that, in order to reject inverted ordering with this argument, an accuracy on the sum of neutrino masses $σ({m_ν})$ of better than 0.02~eV would be required from future cosmological observations.

10 pages, 4 figures, 1 table. Version accepted for publication in Physics of the Dark Universe