Correlation-length exponent of the 3D Anderson-Mott transition with Coulomb interaction
In plain words
For electrons that do not repel, the transition exponent is known numerically to three digits. For real electrons, which repel through the Coulomb force, no reliable value exists.
Precise statement
For 3D disordered electrons with $1/r$ repulsion (Finkelstein nonlinear sigma model; with and without spin-orbit scattering or magnetic field), compute the correlation-length exponent $\nu$ and the dynamical exponent $z$ at the metal-insulator transition with controlled error, and compare with the non-interacting values $\nu \sim 1.57$ (orthogonal), $\sim 1.44$ (unitary) and $\sim 1.37$ (symplectic).
What would settle it
A non-perturbative computation (self-consistent Hartree-Fock with exact diagonalization, functional renormalization group of the sigma model, or quantum Monte Carlo of a lattice model with Coulomb interaction) with finite-size scaling and error bars.
Status in the literature
A density-functional (Kohn-Sham, mean-field) model of a doped semiconductor gave $\nu \sim 1.3$ (Harashima and Slevin, Physical Review B 89, 205108, 2014), suggesting a different universality class; no controlled many-body result exists.