Positronium resonance contribution to the electron g-2

Recently a few authors pointed out that the positroniums give rise to an extra contribution to the electron $g-2$ with the magnitude comparable to the $O(\alpha^5)$ perturbative effect, which is not taken into account in the perturbative calculation up to $O(\alpha^5)$. Here, we scrutinize how the positronium resonances contribute to the electron $g-2$ through the vacuum polarization function, and conclude that there is no additional sizable $O(\alpha^5)$ contribution to the electron $g-2$.