We argue that high-quality data on the reaction $e^+e^-\to ��^+��^-��$ will allow one to determine the doubly-virtual form factor $��\to ��^*��^*$ in a model-independent way with controlled accuracy. This is an important step towards a reliable evaluation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon. When analyzing the existing data for $e^+e^-\to��^+��^-��$ for total energies squared $k^2>1\text{GeV}^2$, we demonstrate that the effect of the $a_2$ meson provides a natural breaking mechanism for the commonly employed factorization ansatz in the doubly-virtual form factor $F_{����^*��^*}(q^2,k^2)$. However, better data are needed to draw firm conclusions.

15 pages, 10 figures; v2: major revision - author list extended, treatment of the left-hand cut corrected, discussion and references updated; v3: version published in EPJC