  Motivated by the excess of the muon content of cosmic ray induced extensive
air showers (EAS), relative to EAS modeling, observed by the Pierre Auger
Observatory, and by the tension between Auger data and air shower simulations
on the maximal muon production depth $X^{\mu}_{\max}$, we investigate the
possibility to modify the corresponding EAS simulation results, within the
Standard Model of particle physics. We start by specifying the kinematic range
for secondary hadron production, which is of relevance for such predictions. We
further investigate the impact on the predicted EAS muon number and on
$X^{\mu}_{\max}$ of various modifications of the treatment of hadronic
interactions, in the framework of the QGSJET-III model, in particular the model
calibration to accelerator data, the amount of the "glue" in the pion, and the
energy dependence of the pion exchange process. None of the considered
modifications of the model allowed us to enhance the EAS muon content by more
than 10\%. On the other hand, for the maximal muon production depth, some of
the studied modifications of particle production give rise up to $\sim 10$
g/cm$^2$ larger $X^{\mu}_{\max}$ values, which increases the difference with
Auger observations.
