Which set of decoherent histories describes the quasi-classical world?
In plain words
The consistent-histories approach assigns probabilities to sequences of events only when the sequences do not interfere with each other. Many incompatible families of such sequences pass this test, and no agreed rule picks the family that matches the classical world.
Precise statement
In the consistent (decoherent) histories formulation, histories are chains of projectors C_alpha = P_alpha_n(t_n) ... P_alpha_1(t_1), and a set is decoherent when the decoherence functional $D(\alpha, \beta) = \operatorname{Tr}[C_\alpha \rho C_\beta^{\mathrm{dag}}]$ is diagonal. Dowker and Kent (J. Stat. Phys. 1996) showed that consistency admits many mutually incompatible sets, including sets quasi-classical up to some time and non-classical afterwards. Find a criterion stated in terms of $H$, $\rho$ and possibly a TPS that selects the quasi-classical set (up to coarse-graining) in standard models, or prove that no criterion using only these data does.
What would settle it
A selection criterion proved to yield the quasi-classical hydrodynamic or pointer-variable set in an explicit model, or an impossibility theorem.