Volume Bounds for Sublevel Sets

It is known that given a bounded pseudoconvex domain $\Omega\subseteq\mathbb{C}^{n}$, there exist $C=C(\Omega)$ and $\alpha=\alpha(\Omega)$ such that for any $u\in PSH(\Omega)$ satisfying $u|_{\partial\Omega}\equiv0$ and $\int_{\Omega}(dd^{c}u)^{n}=1$ (and this is defined), we have

A more precise estimate is

Can one remove the $s^{n-1}$ on the right hand side? Given a compact Kahler manifold $(M,\omega)$ with boundary (and $\omega$ extends smoothly across the boundary), do these estimates generalize if instead one assume $u\in PSH(M,\omega)$, $u|_{\partial M}\equiv0$, and $\int_{\Omega}\omega^{n}=1$? That is, can one bound

One could also ask this question for the complete noncompact case, with appropriately modified assumptions/conclusion.