Can the triple-parton scattering be observed in open charm meson production at the LHC?

We investigate whether the triple-parton scattering effects can be observed in open charm production in proton-proton collisions at the LHC. We use so-called factorized Ansatz for calculations of hard multiple-parton interactions. The numerical results for each parton interaction are obtained within the $k_T$-factorization approach. Predictions for one, two and three $c\bar{c}$ pairs production are given for $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV. Quite large cross sections, of the order of milibarns, for the triple-parton scattering mechanism are obtained. We suggest a measurement of three $D^0$ mesons or three $\overline{D^0}$ antimesons by the LHCb collaboration. Confronting our results with recent LHCb experimental data for single and double $D^0$ (or $\overline{D^0}$) meson production we present our predictions for triple meson final state: $D^0{D}^0{D}^0$ or $\overline{D^0}\overline{D^0}\overline{D^0}$. We present cross sections for the LHCb fiducial volume as well as distributions for $D^0$ meson transverse momentum and three-$D^0$ meson invariant mass. The predicted visible cross sections, including the detector acceptance, hadronization effects and $c\to D^0$ branching fraction, is of the order of a few nanobarns. The counting rates including $D^0\to K^{-}{\pi }^{+}$ branching fractions are given for known or expected integrated luminosities.