  While the low-energy part of the hadronic light-by-light (HLbL) tensor can be
constrained from data using dispersion relations, for a full evaluation of its
contribution to the anomalous magnetic moment of the muon $(g-2)_\mu$ also
mixed- and high-energy regions need to be estimated. Both can be addressed
within the operator product expansion (OPE), either for configurations where
all photon virtualities become large or one of them remains finite. Imposing
such short-distance constraints (SDCs) on the HLbL tensor is thus a major
aspect of a model-independent approach towards HLbL scattering. Here, we focus
on longitudinal SDCs, which concern the amplitudes containing the
pseudoscalar-pole contributions from $\pi^0$, $\eta$, $\eta'$. Since these
conditions cannot be fulfilled by a finite number of pseudoscalar poles, we
consider a tower of excited pseudoscalars, constraining their masses and
transition form factors from Regge theory, the OPE, and phenomenology.
Implementing a matching of the resulting expressions for the HLbL tensor onto
the perturbative QCD quark loop, we are able to further constrain our
calculation and significantly reduce its model dependence. We find that
especially for the $\pi^0$ the corresponding increase of the HLbL contribution
is much smaller than previous prescriptions in the literature would imply.
Overall, we estimate that longitudinal SDCs increase the HLbL contribution by
$\Delta a_\mu^\text{LSDC}=13(6)\times 10^{-11}$. This number does not include
the contribution from the charm quark, for which we find
$a_\mu^{c\text{-quark}}=3(1)\times 10^{-11}$.
