  We investigate the fate of the Roberge-Weiss endpoint transition and its
connection with the restoration of chiral symmetry as the chiral limit of $N_f
= 2+1$ QCD is approached. We adopt a stout staggered discretization on lattices
with $N_t = 4$ sites in the temporal direction; the chiral limit is approached
maintaining a constant physical value of the strange-to-light mass ratio and
exploring three different light quark masses, corresponding to pseudo-Goldstone
pion masses $m_\pi \simeq 100, 70$ and 50 MeV around the transition. A finite
size scaling analysis provides evidence that the transition remains second
order, in the 3D Ising universality class, in all the explored mass range. The
residual chiral symmetry of the staggered action also allows us to investigate
the relation between the Roberge-Weiss endpoint transition and the chiral
restoration transition as the chiral limit is approached: our results,
including the critical scaling of the chiral condensate, are consistent with a
coincidence of the two transitions in the chiral limit; however we are not able
to discern the symmetry controlling the critical behavior, because the critical
indexes relevant to the scaling of the chiral condensate are very close to each
other for the two possible universality classes (3D Ising or O(2)).
