  Low-energy Compton scattering off the proton is used for determination of the
proton polarizabilities. However, the present empirical determinations rely
heavily on the theoretical description(s) of the experimental cross sections in
terms of polarizabilities. The most recent determinations are based on either
the fixed-$t$ dispersion relations (DR) or chiral perturbation theory in the
single-baryon sector ($\chi$PT). The two approaches obtain rather different
results for proton polarizabilities, most notably for $\beta_{M1}$ (magnetic
dipole polarizability). We attempt to resolve this discrepancy by performing a
partial-wave analysis of the world data on proton Compton scattering below
threshold. We find a large sensitivity of the extraction to a few "outliers",
leading us to conclude that the difference between DR and $\chi$PT extraction
is a problem of the experimental database rather than of "model-dependence". We
have specific suggestions for new experiments needed for an efficient
improvement of the database. With the present database, the difference of
proton scalar polarizabilities is constrainedto a rather broad interval:
$\alpha_{E1}-\beta_{M1} = (6.8\, \ldots \, 10.9)\times 10^{-4}$ fm$^3$, with
their sum fixed much more precisely [to $14.0(2)$] by the Baldin sum rule.
