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The $\nu=12/5$ fractional quantum Hall plateau observed in $\mathrm{GaAs}$semiconductor wells is a suspect in the search for non-Abelian Fibonaccianyons. Using the infinite density matrix renormalization group, we find clearevidence 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 Fibonaccianyon, distinguished from its Abelian counterpart by its local quadrupolemoment. However, several experiments at $\nu = 13/5$ observe a re-entrantinteger quantum Hall effect, implying particle-hole symmetry is broken. We ruleout spin polarization as the origin of the asymmetry. Further, we point outextremely close energetic competition between the Read-Rezayi phase and are-entrant integer quantum Hall phase. This competition suggests that evensmall particle-hole symmetry-breaking perturbations can explain theexperimentally observed asymmetry between $\nu = 12/5$ and $13/5$. We find thatat $\nu=12/5$ Landau level mixing favors the Read-Rezayi phase over there-entrant phase.

Fibonacci anyons and charge density order in the 12/5 and 13/5 plateaus