  High-statistics data on the $e^+e^-\to \pi^+\pi^-$ cross section and the pion
vector form factor have been obtained recently by several collaborations.
Unfortunately, there are some tensions between different datasets, especially
the most precise ones, which have not been resolved so far. Additional
independent constraints on the data are therefore of interest. We consider a
parametrization-free method of analytic extrapolation proposed recently, which
is based on a mixed phase and modulus extremal problem and combines rigorous
upper and lower bounds with numerical simulations to account for the
statistical distributions of the input and output values. Spacelike data on the
form factor and measurements of the modulus in the region $(0.65-0.71)$ GeV are
used as input. In previous works, the formalism was applied for extrapolating
the form factor to low energies. In the present work, we use it as a stringent
and model-independent test of consistency with analyticity and unitarity for
the high-statistics data around the $\rho$ resonance. The study reveals some
inconsistencies, in particular below the $\rho$ peak the BABAR data are
slightly higher than the band of extrapolated values, while above the $\rho$
peak all the data are situated at the lower edge of the band. The implications
of the results on the two-pion vacuuum polarization contribution to the
anomalous magnetic moment of the muon are briefly discussed.
