Construct the physical Hilbert space of the Wheeler-DeWitt equation
In plain words
The Wheeler-DeWitt equation is the quantum version of Einstein's equations for the whole universe. There is no rigorous way to normalize its solutions, so probabilities are not defined.
Precise statement
For $3+1$ general relativity (vacuum or with a scalar field) quantized canonically, construct a physical inner product on solutions of the Hamiltonian and momentum constraints $H \Psi = 0, H_{i} \Psi = 0$, with a complete set of Dirac observables represented as self-adjoint operators, beyond minisuperspace truncations. An answer is a rigorous construction (for example group averaging with proven convergence) or a proof that none exists for a stated operator ordering and regularization.
What would settle it
A mathematically rigorous construction of the physical Hilbert space for a field-theoretic (not minisuperspace) gravitational model in 3+1 dimensions.