COSMO
In the literature: open
Cosmological measurement of the neutrino mass sum to 0.015 eV
In plain words
Cosmology can weigh neutrinos because heavier neutrinos smooth out the clumping of matter. The task is to measure the total mass with cosmological data, not just bound it.
Precise statement
Measure sum $m_\nu$ with uncertainty $\sigma \le 0.015\,\mathrm{eV}$ from CMB lensing, galaxy clustering and BAO, with the reionization optical depth $\tau$ measured independently to $\sigma(\tau) \sim 0.002$. An answer is a value that also discriminates normal from inverted ordering.
What would settle it
Simons Observatory and LiteBIRD-era CMB lensing and large-scale polarization combined with final DESI and Euclid.