  We theoretically propose a possibility of realizing high $T_c$
superconductivity having $s\pm$-wave symmetry in the heavily hole-doped
infinite layer nickelates La$_{1-x}$Sr$_x$NiO$_2$. We consider situations where
the original $P4/mmm$ symmetry of LaNiO$_2$ is maintained even for a
significant amount of Sr substitution by growing thin films on substrates
having tetragonal symmetry. Considering such cases is indeed justified by our
phonon calculations. For electron configurations somewhat close to $d^8$, the
interaction between the $d_{x^2-y^2}$ band and the other $3d$ bands that lie
just below the Fermi level results in an enhancement of superconductivity where
the sign of the gap function is reversed between the former and the latter
bands. The strong enhancement of superconductivity can be attributed to the
large energy level offset between $d_{x^2-y^2}$ and other orbitals due to the
absence of the apical oxygens, as has been pointed out in previous studies.
