let $(X,g,\omega)$ be a non compact complete K\"ahler manifold of dimension $m\geq3$. Let $\nabla$ the covariant derivative wrt $g$ and $Riem$ the curvature tensor of $g$. Suppose that \begin{equation} |\nabla^{n}Riem|_{g}\leq C_{n} \qquad n\in\mathbb{N} \textrm{ and }C_{n}>0 \end{equation} i.e. the curvature tensor and all of its derivatives are uniformly bounded. Let moreover $\mathcal{L}$ be a holomorphic line bundle over $X$. Let $K\subset X$ be a compact analytic subvariety of dimension $m-1$ and $\sigma$ be a section of $\mathcal{L}|_{X\setminus K}$ such that $\sigma\neq 0$ everywhere.

Is $\sigma$ extendable to a holomorphic section of $\mathcal{L}$?