Can Majorana zero modes in quantum Hall edges survive edge reconstruction?

Parafermion zero modes can be trapped in the domain walls of quantum Hall edges proximitized by superconductors and ferromagnets. The $\nu =1\mathrm{/}3$ fractional quantum Hall side strip arising due to edge reconstruction of a $\nu =1$ edge doubles the number of topological sectors such that each of them is $Z_2\times Z_2$ degenerate. The many-body spectrum displays a $4\pi$ Josephson periodicity, with the states in each $Z_2$ being energetically decoupled. Signatures of the new states appear in the fractional Josephson current when the edge velocities are taken to be different.