  In the context of the ongoing search for the QCD critical point at the
Relativistic Heavy-Ion Collider, we study the equation of state near the
critical point in the temperature and baryon chemical potential plane. We use
the parametric representation introduced in earlier literature, which maps the
universal 3D Ising equation of state onto the QCD phase diagram using several
non-universal parameters. We focus on the quartic cumulant of the baryon
number, or baryon number susceptibility~$\chi_4^B$, which can be accessed
experimentally via net-proton fluctuation kurtosis measurements. It was
originally predicted, through universality arguments based on the {\em leading}
singular contribution, that $\chi_4^B$ and net-proton kurtosis should show a
specific non-monotonic behavior due to the critical point. In particular, when
following the freeze-out curve on the phase diagram by decreasing beam energy,
the kurtosis is expected to dip, and then peak, when the beam energy scan
passes close to the critical point. We study the effects of the non-universal
and thus far unknown parameters of the Ising-to-QCD mapping on the behavior
of~$\chi_4^B$. We find that, while the peak remains a solid feature, the
presence of the critical point does not necessarily cause a dip in $\chi_4^B$
on the freezeout line {\em below} the transition temperature. The critical
point contribution to the dip appears only for a narrow set of mapping
parameters, when subleading singular terms are sufficiently suppressed.
