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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~$χ_4^B$, 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 $χ_4^B$ 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~$χ_4^B$. We find that, while the peak remains a solid feature, the presence of the critical point does not necessarily cause a dip in $χ_4^B$ 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.