Uniform Sobolev inequality for blowups of compact Kähler varieties

Let $(X,\omega)$ be a compact Kähler variety of dimension $n$. Let $p\colon Y\to X$ be a blowup at a smooth center, and let $\eta$ be a Kähler form on $Y$. For $0<\epsilon\ll1$, define

$$\omega_\epsilon=r^*\omega+\epsilon\eta.$$

Uniform Sobolev inequality conjecture. The metrics $(Y,\omega_\epsilon)$ have a uniform Sobolev constant for $0<\epsilon\leqslant1$: there exists a constant $C_S$, independent of $\epsilon$, such that every nonnegative $\mathcal C^1$ function $f$ compactly supported in the smooth locus of $Y$ satisfies

$$\left(\int_Y |f|^{\frac{2n}{n-1}}\omega_\epsilon^n\right)^{\frac{n-1}{n}} \leqslant C_S\int_Y\left(\lVert\nabla_{\omega_\epsilon}f\rVert^2+|f|^2\right)\omega_\epsilon^n.$$

Such an estimate is needed to adapt the Bando–Siu method for Hermitian–Yang–Mills metrics to singular compact Kähler varieties; the paper presents it as the global uniform Sobolev inequality required for the analytic argument, without resolving it.