Extension of holomorphic principal bundles characterized by vanishing Atiyah class

Let $M$ be a domain in a Stein manifold, let $\hat{M}$ be its envelope of holomorphy, and let $E \to M$ be a holomorphic principal bundle. The extension conjecture. The bundle $E$ extends uniquely to a holomorphic principal bundle over $\hat{M}$ if and only if its Atiyah class vanishes. Since holomorphic principal bundles on Stein manifolds admit holomorphic connections, the vanishing of the Atiyah class is necessary for extension; the converse is the proposed extension criterion.