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After the first predictions in seventies that in dense nuclear matter, for example, in neutron stars there could be pion condensation phenomenon, it was shown that at least s-wave pion condensation is unlikely to occur in such a medium in different approaches including NJL model consideration with neutrality and $β$-equilibrium condition. Then it has been found a condition that can promote this phenomenon in dense baryonic matter, it is chiral imbalance, so now it is interesting if this generation can survive rather strict requirement of electric neutrality and $β$-equilibrium. Here in this paper the generation of pion condensation in dense baryonic matter by chiral imbalance is investigated. It is shown that electric neutrality and $β$-equilibrium conditions do not spoil this phenomenon and this leads to interesting applications for physics of neutron stars. There are several possible mechanisms of generation of chiral imbalance in dense matter. In a view of latest and forthcoming NICER results and the first observed and possibly new events of neutron star mergers it is rather interesting to explore how possible chiral imbalance in neutron star can influence and change EOS and, hence, mass-radius relation. It is a first step in that direction.