Anyons are particles which are neither bosons nor fermions, setting them apart from all known particles.  The ν = 12/5 fractional quantum Hall plateau observed in GaAs wells is a suspect in the search for non-Abelian Fibonacci anyons.  Fibonacci anyons are special in that they are capable of performing universal topological quantum computation.  Using the infinite density matrix renormalization group, we find clear evidence that—in the absence of Landau level mixing—fillings ν = 12/5 and ν = 13/5 are in the k = 3 Read-Rezayi phase, and thus supports Fibonacci anyons.  We also find an extremely close energetic competition between the Read-Rezayi phase and a charge-density ordered phase, and provide a mechanism explaining the observed differences between the two states in experiments.