  It has been predicted that the phase sensitive part of the current through a
non-abelian $\nu = 5/2$ quantum Hall Fabry-Perot interferometer will depend on
the number of localized charged $e/4$ quasiparticles (QPs) inside the
interferometer cell. In the limit where all QPs are far from the edge, the
leading contribution to the interference current is predicted to be absent if
the number of enclosed QPs is odd and present otherwise, as a consequence of
the non-abelian QP statistics. The situation is more complicated, however, if a
localized QP is close enough to the boundary so that it can exchange a Majorana
fermion with the edge via a tunneling process. Here, we derive an exact
solution for the dependence of the interference current on the coupling
strength for this tunneling process, and confirm a previous prediction that for
sufficiently strong coupling, the localized QP is effectively incorporated in
the edge and no longer affects the interference pattern. We confirm that the
dimensionless coupling strength can be tuned by the source-drain voltage, and
we find that not only does the magnitude of the even-odd effect change with the
strength of bulk-edge coupling, but in addition, there is a universal shift in
the interference phase as a function of coupling strength. Some implications
for experiments are discussed at the end.
