  The anomalous $4\pi$-periodic ac Josephson effect, a hallmark of topological
Josephson junctions, was experimentally observed in a quantum spin Hall
insulator. This finding is unexpected due to time-reversal symmetry preventing
the backscattering of the helical edge states and therefore suppressing the
$4\pi$-periodic component of the Josephson current. Here we analyze the
two-particle inelastic scattering as a possible explanation for this
experimental finding. We show that a sufficiently strong inelastic scattering
restores the $4\pi$-periodic component of the current beyond the short
Josephson junction regime. Its signature is an observable peak in the power
spectrum of the junction at half the Josephson frequency. We propose to use the
exponential dependence of the peak width on the applied bias and the magnitude
of the dc current as means of verifying that the inelastic scattering is indeed
the mechanism responsible for the $4\pi$-periodic signal.
