CP violation in the rare Higgs decays via exchange of on-shell almost degenerate Majorana neutrinos, $H \to \nu_k N_j \to \nu_k \ell^{-} U {\bar D}$ and $H \to \nu_k N_j \to \nu_k \ell^{+} {\bar U} D$

We investigate rare decays of Higgs via exchange of two almost degenerate heavy on-shell Majorana neutrinos $N_j$ ($j=1,2$): $\Gamma_{\pm} = \Gamma(h \to \nu_k N_j \to \nu_k \ell^{\pm} \pi^{\mp})$, and into the open quark channels $\Gamma_{\pm} = \Gamma(h \to \nu_k N_j \to \nu_k \ell^{\pm} U D)$, where $U D$ are two jets of open quarks (${\bar U} D$, or $U {\bar D}$, where $U=u, c$ and $D=d, s$). The related CP violation asymmetry $A_{\rm CP} = (\Gamma_{-} - \Gamma_{+})/(\Gamma_{-} + \Gamma_{+})$ is studied in detail. We take into account the $N_1$-$N_2$ overlap and oscillation effects. We can see that for certain, presently acceptable, range of input parameters, such decays with open quark channels, and their asymmetries, could be detected in the International Linear Collider (ILC).