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

Majorana and parafermion zero modes can be trapped at domain walls of quantum hall edges proximitised by

superconductors and ferromagnets. Softening of the edge potential of the ν=1 quantum Hall edge  allows for a narrow  ν=1/3

strip to be deposited near the edge by reconstruction. Subsequent edge renormalisation due to inter-edge interactions and 

disorder induced tunneling leads to a new edge structure, which naively allows for both Majorana and parafermion zero modes.

However,  constraints imposed by the total charge and spin of the bulk quantum Hall systems show that the ground state manifold

of the reconstructed system has a four-fold degeneracy  with two even parity and two odd parity sectors, thus leading 

to a Z₂ Χ Z₂ symmetry.  When the ferromagnet between

the superconductors is removed, the many-body spectrum shows a 4π Josephson effect with the states in each Z₂ being

energetically decoupled. Signatures of the new states appear when the edge velocities of the reconstructed edges are different.