Generalized Gauduchon metrics on compact complex spaces

Let $X$ be an irreducible reduced compact complex space of dimension $n$, and let $\omega$ be a Hermitian metric on $X$. For each $1\le k\le n-1$, we prove that there exist a constant $c_k$ and a smooth positive function $\rho$ on $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ such that $\text{ii}\mathrm{\partial }\bar{\mathrm{\partial }}(\rho {\omega }^k)\wedge {\omega }^{n-k-1}=c_k\rho {\omega }^n$. The function $\rho$ is bounded above and bounded away from zero. The constant $c_k$ is unique, and $\rho$ is unique up to a positive multiple among bounded smooth positive solutions. We also prove that the full sign set $\{-1,0,1\}$ of the generalized Gauduchon constants $c_k$ on a compact complex manifold is preserved by finite sequences of blow-ups with smooth centers.