  The phase diagram of dense quark matter with chiral imbalance is considered
with the conditions of electric neutrality and $\beta$-equilibrium. It has been
shown recently that chiral imbalance can generate charged pion condensation in
dense quark matter, so it was interesting to verify that this phenomenon takes
place in realistic physical scenarios such as electrically neutral matter in
$\beta$-equilibrium, because a window of pion condensation at dense quark
matter phase diagram (without chiral imbalance) predicted earlier was closed by
the consideration of these conditions at the physical current quark mass. In
this paper it has been shown that the charged pion condensation is generated by
chiral imbalance in the dense electric neutral quark/baryonic matter in
$\beta$-equilibrium, i. e. matter in neutron stars. It has been also
demonstrated that pion condensation is inevitable phenomenon in dense quark
matter with chiral imbalance if there is non-zero chiral imbalance in two
forms, chiral and chiral isospin one. It seems that in this case pion
condensation phase can be hardly avoided by any physical constraint on isopin
imbalance and that this conclusion can be probably generalized from neutron
star matter to the matter produced in heavy ion collisions or in neutron star
mergers. The chiral limit and the physical piont (physical pion mass) has been
considered and it was shown that the appearance of pion condensation is not
much affected by the consideration of non-zero current quark mass.
