Local uniform Sobolev inequality near an isolated analytic singularity

Let $X\subseteq\mathbb C^n\times\mathbb C^k=N$ be an $m$-dimensional integral complex analytic subvariety defined near the origin $o$. Let $(\mathbf z,\mathbf w)$ be coordinates on $\mathbb C^n\times\mathbb C^k$, and for $0\leqslant\delta\leqslant1$ define

$$\omega_\delta=\frac{\sqrt{-1}}{2}\partial\overline{\partial}\left(|\mathbf z|^2+\delta^2\log(|\mathbf z|^2)+|\mathbf w|^2\right).$$

Local uniform Sobolev inequality conjecture. There exist a neighbourhood $U$ of $o$ in $N$ and a constant $C$, independent of $\delta$, such that every positive $\mathcal C^1$ function compactly supported in the smooth locus of $U\cap X\setminus\{o\}$ satisfies

$$\int_X h\,\omega_\delta^m\leqslant C\int_X|\nabla_{X_\delta}h|\,\omega_\delta^m.$$

This is the local version used to reduce the global uniform Sobolev inequality to a neighbourhood of the singularity. Its resolution is not supplied in the paper.