Critical exponents of the quantum Hall transition with Coulomb interaction
In plain words
Real quantum Hall transitions involve repelling electrons, which changes how time and length scale near the transition. Experiments measure a combined exponent of about 0.42, but theory has not computed it with the repulsion included.
Precise statement
AlGaAs/GaAs plateau transitions give $\kappa=1/(z\nu)\sim 0.42$ (Li et al. 2009), with $z\sim 1$ as expected for $1/r$ interaction. Short-range interactions are irrelevant at the non-interacting fixed point, while $1/r$ Coulomb interaction is relevant. Compute $\nu\ \text{and}\ z$ at the interacting fixed point and decide whether $\nu$ equals the non-interacting value.
What would settle it
A controlled calculation (Hartree-Fock network models, sigma-model renormalization, or exact methods on large interacting systems) giving $\nu\text{ and }z$ with error bars, compared with temperature- and frequency-scaling experiments.