  We study the connection between absolute neutrino mass and neutrino mixing
parameters within $SO(10)$-inspired leptogenesis. We show that current favoured
values of the unknown neutrino mixing parameters point toward values of the
absolute neutrino mass scale that will be fully tested by cosmological
observations and neutrinoless double beta decay experiments during next years.
In particular, for $m_{D2}/m_{\rm charm} \leq 5$, where $m_{D2}$ is the
intermediate Dirac neutrino mass, and for current best fit values of the Dirac
phase $\delta$ and the atmospheric neutrino mixing angle $\theta_{23}$, we
derive a lower bound on the neutrinoless double beta decay effective neutrino
mass $m_{ee} \gtrsim 31\,{\rm meV}$ and on the sum of the neutrino masses
$\sum_i m_i \gtrsim 125\,{\rm meV}$. These lower bounds hold for normally
ordered neutrino masses, as currently favoured by global analyses and
approximately for $\delta \in [155^\circ,240^\circ]$ and $\theta_{23}$ in the
second octant. If values in this region will be confirmed by future planned
long baseline experiments, then a signal at next generation neutrinoless double
beta decay experiments is expected, despite neutrino masses being normally
ordered. Outside the region, the lower bounds strongly relax but a great
fraction of the allowed range of values still allows a measurement of the
lightest neutrino mass. Therefore, in the the next years low energy neutrino
experiments will provide a very stringent test of $SO(10)$-inspired
leptogenesis, resulting either in severe constraints or in a strong evidence.
