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We study the impact of the mixing (LR mixing) between the standard model $W$ boson and its hypothetical, heavier right-handed parter $W_R$ on the neutrinoless double beta decay ($0νββ$-decay) rate. Our study is done in the minimal left-right symmetric model assuming type-II dominance scenario with charge conjugation as the left-right symmetry. We then show that the $0νββ$-decay rate may be dominated by the contribution proportional to this LR mixing, which at the hadronic level induces the leading-order contribution to the interaction between two pions and two charged leptons. The resulting long-range pion exchange contribution can significantly enhance the decay rate compared to previously considered short-range contributions. Finally, we find that even if future cosmological experiments rule out the inverted hierarchy for neutrino masses, there are still good prospects for a positive signal in the next generation of $0νββ$-decay experiments.