  With the long-standing tension between experiment and Standard-Model (SM)
prediction in the anomalous magnetic moment of the muon, $a_\mu=(g-2)_\mu/2$,
at the level of 3-4$\sigma$, it is natural to ask if there could be a sizable
effect in the electric dipole moment (EDM) $d_\mu$ as well. In this context it
has often been argued that in UV complete models the electron EDM, which is
very precisely measured, excludes a large effect in $d_\mu$. However, the
recently observed 2.5$\sigma$ tension in $a_e=(g-2)_e/2$, if confirmed,
requires that the muon and electron sectors effectively decouple to avoid
constraints from $\mu\to e\gamma$. We briefly discuss UV complete models that
possess such a decoupling, which can be enforced by an Abelian flavor symmetry
$L_\mu-L_\tau$. We show that, in such scenarios, there is no reason to expect a
correlation between the electron and muon EDM, so that the latter can be
sizable. New limits on $d_\mu$ improved by up to two orders of magnitude are
expected from the upcoming $(g-2)_\mu$ experiments at Fermilab and J-PARC.
Beyond, a proposed dedicated muon EDM experiment at PSI could further advance
the limit. In this way, future improved measurements of $a_e$, $a_\mu$, as well
as the fine-structure constant $\alpha$ are not only set to provide exciting
precision tests of the SM, but, in combination with EDMs, to reveal crucial
insights into the flavor structure of physics beyond the SM.
