  We perform a dispersive analysis of the $\omega\pi$ electromagnetic
transition form factor, using as input the discontinuity provided by unitarity
below the $\omega\pi$ threshold and including for the first time experimental
data on the modulus measured from $e^+e^-\to\omega\pi^0$ at higher energies.
The input leads to stringent parameterization-free constraints on the modulus
of the form factor below the $\omega\pi$ threshold, which are in disagreement
with some experimental values measured from $\omega\to \pi^0\gamma^*$ decay. We
discuss the dependence on the input parameters in the unitarity relation, using
for illustration an $N/D$ formalism for the P partial wave of the scattering
process $\omega\pi \to \pi\pi$, improved by a simple prescription which
simulates the rescattering in the crossed channels. Our results confirm the
existence of a conflict between experimental data and theoretical calculations
of the $\omega\pi$ form factor in the region around 0.6 GeV and bring further
arguments in support of renewed experimental efforts to measure more precisely
the $\omega\to\pi^0\gamma^*$ decay.
