Holomorphic maps between moduli spaces II

We prove that forgetful maps are the only non-constant holomorphic maps ${\mathcal{M}}_{g,r}\to {\mathcal{M}}_{g^{\mathrm{\prime }},r^{\mathrm{\prime }}}$ between moduli spaces, as long as $g\ge 4$ and $g^{\mathrm{\prime }}\le 3\cdot 2^{g-3}$.