  The $\nu=12/5$ fractional quantum Hall plateau observed in $\mathrm{GaAs}$
semiconductor wells is a suspect in the search for non-Abelian Fibonacci
anyons. Using the infinite density matrix renormalization group, we find clear
evidence that fillings $\nu = 12/5$ and $\nu=13/5$ are in the $k = 3$
Read-Rezayi phase in the absence of particle-hole symmetry-breaking effects.
The lowest energy charged excitation is identified as a non-Abelian Fibonacci
anyon, distinguished from its Abelian counterpart by its local quadrupole
moment. However, several experiments at $\nu = 13/5$ observe a re-entrant
integer quantum Hall effect, implying particle-hole symmetry is broken. We rule
out spin polarization as the origin of the asymmetry. Further, we point out
extremely close energetic competition between the Read-Rezayi phase and a
re-entrant integer quantum Hall phase. This competition suggests that even
small particle-hole symmetry-breaking perturbations can explain the
experimentally observed asymmetry between $\nu = 12/5$ and $13/5$. We find that
at $\nu=12/5$ Landau level mixing favors the Read-Rezayi phase over the
re-entrant phase.
