Assume $X$ is a homogeneous manifold. If $\mu=f\omega^{n}$ is a probability measure with density $0\leq f\in L^{p}(\omega^{n})$, $p>1$, then the unique solution $\varphi\in PSH(X,\omega)\in{\mathcal{C}}^{0}(X)$ to the normalized Monge-Ampère equation

$$(\omega+dd^{c}\varphi)^{n}=\mu=f\omega^{n},\,\sup_{X}\varphi=-1,$$

is Hölder continuous of exponent $\gamma>0$, for all $\gamma<2/(2+nq)$, where $q=p/(p-1)$ is the conjugate exponent to $p$.