  The production of the $K_S^0$ meson in high multiplicity $pp$ collisions at
$\sqrt{s} =$ 13 TeV is investigated considering the hybrid formalism and the
solution of the running coupling Balitsky - Kovchegov equation. The associated
cross section is estimated and compared with the experimental data for the
transverse momentum spectrum. Moreover, we analyze the self-normalized yields
of $K_S^0$ mesons as a function of the multiplicity of coproduced charged
hadrons and demonstrate that a steep increasing is theoretically predicted. A
comparison with the ALICE data is presented considering two distinct solutions
of the BK equation.
