  We show that successful strong thermal leptogenesis, where the final
asymmetry is independent of the initial conditions and in particular a large
pre-existing asymmetry is efficiently washed-out, favours values of the
lightest neutrino mass $m_1 \gtrsim 10\,{\rm meV}$ for normal ordering (NO) and
$m_1 \gtrsim 3\,{\rm meV}$ for inverted ordering (IO) for models with
orthogonal matrix entries respecting $|\Omega_{ij}^2| \lesssim 2$. . We show
analytically why lower values of $m_1$ require a high level of fine tuning in
the seesaw formula and/or in the flavoured decay parameters (in the electronic
for NO, in the muonic for IO). We also show how this constraint exists thanks
to the measured values of the neutrino mixing angles and can be tighten by a
future determination of the Dirac phase. Our analysis also allows to place more
stringent constraint for a specific model or class of models, such as
$SO(10)$-inspired models, and shows that some models cannot realise strong
thermal leptogenesis for any value of $m_1$. A scatter plot analysis fully
supports the analytical results. We also briefly discuss the interplay with
absolute neutrino mass scale experiments concluding that they will be able in
the coming years to either corner strong thermal leptogenesis or find positive
signals pointing to a non-vanishing $m_1$. Since the constraint is much
stronger for NO than for IO, it is very important that new data from planned
neutrino oscillation experiments will be able to solve the ambiguity.
