  The hardness of the energy spectrum of neutral pions produced in proton-air
interactions at ultra-high energies, above $10^{18}$ eV, is constrained by the
steepness of the shower-to-shower distribution of the number of muons in
muon-depleted extensive air showers. In this work, we find that this steepness,
quantified by the parameter $\Lambda_\mu$, evolves with the depth of the shower
maximum, $X_{\max}$, assuming a universal value for shallow showers and an
enhanced dependence on the high-energy hadronic interaction model for deep
showers. We show that Xmax probes the so-called hadronic activity of the first
interaction, thus allowing direct access to the energy spectrum of neutral
pions in different regions of the kinematic phase space of the first
interaction. We verify that the unbiased measurement of $\Lambda_\mu$ is
possible for realistic mass composition expectations. Finally, we infer that
the statistical precision in $\Lambda_\mu$ required to distinguish between
hadronic interaction models can be achieved in current extensive air shower
detectors, given their resolution and exposure.
