  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 $g-2$.
