CM
In the literature: open
Is the $12/5$ quantum Hall state a Fibonacci anyon phase
In plain words
The $12/5$ state in gallium arsenide may host Fibonacci anyons, whose braiding alone could perform any quantum computation, but this has not been tested.
Precise statement
For the $\nu = 12/5$ state in GaAs, determine whether the topological order is the particle-hole conjugate of the Read-Rezayi $Z_3$ parafermion state (Fibonacci anyons, $e* = e/5$) or an Abelian hierarchy state; the candidates differ in $\kappa_{xy}$ and in tunnelling exponents. An answer is $\kappa_{xy}$ and $e*$ measurements consistent with one candidate.
What would settle it
Thermal Hall conductance at $12/5$ plus shot-noise quasiparticle charge measurement.