Continuity conjecture for restriction of extensions

Let $S^G$ be a Yang–Mills-type theory, let $\imath:G\hookrightarrow\hat{G}$ be a basic extension of $G$, and suppose that $\hat{P}$ is compact. For a ring $R$ and a group $\mathbb{G}$, the extension spaces associated with the extended gauge group and with $\Omega^1_{eq}(\hat{P};\hat{\mathfrak{g}})$ are equipped with the described topologies, and $\imath^*$ denotes the corresponding map.

Continuity conjecture. For every such $S^G$, $\imath$, $R$, and $\mathbb{G}$, the map $\imath^*$ is continuous in the described topologies.

If true, this would make $\imath^*$ a morphism of $R[\mathbb{G}]$-module bundles, providing the topological compatibility needed for the bundle construction. The source supplies no proof of continuity.