Descent for G-bundles over Spec B_dR in the inscribed setting

Establish descent for principal G-bundles over Spec \mathbb{B}_{dR} in the inscribed context; specifically, prove that the fibered category (Spec \mathbb{B}_{dR})^*\mathcal{B} of G-bundles is a v-stack by showing effective descent holds for such bundles. Equivalently, verify that the natural prestack of G-bundles over Spec \mathbb{B}_{dR} satisfies v-descent.

Background

In defining the \mathbb{B}^+_{dR}-affine Grassmannian, the paper requires a robust stack formalism for G-bundles over period rings. While the inscribed pullback over Spec \mathbb{B}^+_{dR} is a v-stack, the analogous construction over Spec \mathbb{B}_{dR} is only shown to be a prestack because the authors do not yet have descent.

Proving descent in this context would upgrade the prestack to a v-stack, streamlining the foundations of the inscribed \mathbb{B}^+{dR}/\mathbb{B}{dR} affine Grassmannian and potentially simplifying several constructions and arguments relying on moduli of bundles over these period rings.