  Compared to $\mu \to e \gamma$ and $\mu \to e e e$, the process $\mu \to e$
conversion in nuclei receives enhanced contributions from Higgs-induced lepton
flavor violation. Upcoming $\mu \to e$ conversion experiments with drastically
increased sensitivity will be able to put extremely stringent bounds on
Higgs-mediated $\mu \to e$ transitions. We point out that the theoretical
uncertainties associated with these Higgs effects, encoded in the couplings of
quark scalar operators to the nucleon, can be accurately assessed using our
recently developed approach based on $SU(2)$ Chiral Perturbation Theory that
cleanly separates two- and three-flavor observables. We emphasize that with
input from lattice QCD for the coupling to strangeness $f_s^N$, hadronic
uncertainties are appreciably reduced compared to the traditional approach
where $f_s^N$ is determined from the pion--nucleon $\sigma$-term by means of an
$SU(3)$ relation. We illustrate this point by considering Higgs-mediated lepton
flavor violation in the Standard Model supplemented with higher-dimensional
operators, the two-Higgs-doublet model with generic Yukawa couplings, and the
Minimal Supersymmetric Standard Model. Furthermore, we compare bounds from
present and future $\mu \to e$ conversion and $\mu \to e \gamma$ experiments.
