  The helical edge states of time-reversal invariant two-dimensional
topological insulators are protected against backscattering in idealized
models. In more realistic scenarios with a shallow confining potential at the
sample boundary, additional strongly interacting edge states may arise, that
could interfere with the topological protection of edge conduction. We find
that interaction effects within the reconstructed edges are well described by
the Luttinger liquid model. While interactions between this Luttinger liquid
and the helical edge states can in principle give rise to dynamical spin
polarization and the breaking of time-reversal symmetry, we demonstrate that
random spin-orbit coupling strongly suppresses such dynamical spin
polarization, resulting in the persistence of near quantized edge conduction.
