Extension conjecture for holomorphic first order structures with trivial kernel

Let $M$ be a domain in a Stein manifold, let $\hat{M}$ be its envelope of holomorphy, and let a holomorphic first order structure on $M$ have an underlying holomorphic principal bundle with trivial kernel. The first order structure extension conjecture. The underlying holomorphic principal bundle extends to a holomorphic principal bundle on $\hat{M}$ if and only if the first order structure extends to $\hat{M}$. The surrounding results establish extension across codimension at least two for the underlying bundle, while the stated equivalence for envelopes of holomorphy remains the proposed claim.