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We investigate rare decays of tau leptons that occur via exchange of heavyon-shell neutrinos $N_j$ ($j=1,2$). These neutrinos can be either Dirac orMajorana, and are considered to be almost degenerate in mass. The decays canthus be either lepton number conserving (LNC), $\tau^{\mp} \to \pi^{\mp} N_j\to \pi^{\mp} \mu^{\mp} \pi^{\pm}$, or lepton number violating (LNV),$\tau^{\mp} \to \pi^{\mp} N_j \to \pi^{\mp} \mu^{\pm} \pi^{\mp}$. If neutrinosare Dirac, only LNC decays are possible. If they are Majorana, both LNC and LNVare possible. We derive the corresponding expressions for the effective decaywidths $\Gamma_{{\rm eff},\mp}^{\rm (X)}$ (X=LNC, LNV) of these rare decays,where we account for $N_1$-$N_2$ overlap and oscillation effects and for thefinite detector length effects. We then numerically evaluate these decay widthsas well as the related CP violation asymmetry width $\Delta \Gamma_{\rmCP}^{\rm (X)} = (\Gamma_{{\rm eff},-}^{\rm (X)} - \Gamma_{{\rm eff},+}^{\rm(X)})$. We conclude that for certain, presently allowed, ranges of theheavy-light neutrino mixing parameters, such decays and asymmetries could beobserved in Belle II experiment.

Rare tau decays via exchange of on-shell almost degenerate Majorana  neutrinos, $\tau^{\mp} \to \pi^{\mp} N_j \to \pi^{\mp} \mu^{\mp} \pi^{\pm}$  and $\tau^{\mp} \to \pi^{\mp} N_j \to \pi^{\mp} \mu^{\pm} \pi^{\mp}$