  The $\nu_M{\rm SM}$ is defined as the SM extended to include dimension-5
operators. In this model neutrino masses violate lepton number, and two
parameters of the lepton mixing matrix, the Majorana phases, are yet to be
constrained. One combination of these phases and the neutrino masses, often
denoted by $m_{ee}$, is probed by neutrinoless double beta decays
($0\nu\beta\beta$). We explore what information may be obtained beyond
$0\nu\beta\beta$, and how it depends on the lightest neutrino mass. We point
out that with current central values of the mixing parameters, $\Delta L_e=2$
and $\Delta L_e=\Delta L_\mu = 1$ (or $\Delta L_e=2$ and $\Delta L_\mu = 2$)
processes cannot simultaneously vanish, providing a no-lose theorem, in
principle, for excluding the $\nu_M{\rm SM}$, even in the case of normal mass
ordering.
