Uniform Bounds in Families

Problem 5. Let $X$ be a connected component of the moduli space of Kahler-Einstein metrics on a compact complex manifold Y of general type, and let $\pi:\mathcal{X}\to X$ be the corresponding universal family. Then there exists $N\in\mathbb{N}$ and a closed embedding $\iota:\mathcal{X}\hookrightarrow X\times\mathbb{P}^{N}$ over $X$ using powers of the relative canonical bundle $K_{\mathcal{X}/X}^{\ell}$ for sufficiently large $\ell\in\mathbb{N}$. Let $\omega_{FS}$ be the Fubini-Study metric on $\mathbb{P}^{N}$, so that we can write any Kahler-Einstein metric $\omega_{KE}\in X$ (corresponding to some $z\in X$) as $\omega_{KE}=\frac{1}{\ell}\iota_{z}^{\ast}\omega_{FS}+\sqrt{-1}\partial\overline{\partial}u$,where $\iota_{z}:\pi^{-1}(z)\hookrightarrow\{z\}\times\mathbb{P}^{N}$ is the restriction of $\iota$, and $u\in PSH(Y,\frac{1}{\ell}\iota_{z}^{\ast}\omega_{FS})\cap C^{\infty}(Y)$.

Do there exist constants $C=C(X)$ and $\alpha=\alpha(X)>0$ such that $\text{Vol}_{\omega_{FS}}(\{u<-s\})\leq Ce^{-\alpha s}$ for all $\omega_{KE}\in X$?