As long as there is no reason to prefer one of these Lagrangians (and
there was none in the 1950s), this means complete equivalence.
The standard model predicts L_W quantitatively - independent of
measured neutrino properties. From this the Lagrangian of
a Majorana neutrino with SM interactions "L_M(SM)"
can be determined (but the fact that it can, does not
automatically prove equivalence).
The manuscript shows that L_M(SM) is different from L_M(equivalent).
Weyl neutrinos can have Majorana masses. Therefore the manuscript's
conclusion "If the SM is quantitatively valid, neutrinos are Weyl fermions",
does not exclude fermion-number violating processes.