Convexity properties of coverings of 1-convex manifolds

We show the following result: let X be a 1-convex manifold, A its exceptional set, k = dimA and p : Y → X any covering. Then Y can be exhausted by an increasing sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Y_{\nu}\}_{\nu \in \mathbb{N}}$$\end{document} of smoothly bounded k-convex domains.