Projective-injective splitting conjecture for extension modules

Let $S^G$ be a Yang–Mills-type theory and let $\imath:G\hookrightarrow\hat{G}$ be a basic extension of $G$ with $\hat{P}$ compact. Let $R$ be a ring and $\mathbb{G}$ a group. The extension space $\operatorname{Ext}(S^G;\Omega^1_{eq}(\hat{P};\hat{\mathfrak{g}}))$ is considered as an $R[\mathbb{G}]$-module, as are the spaces $\operatorname{Ext}(S^G;\widehat{\operatorname{Conn}}(\hat{P};\hat{\mathfrak{g}}))$.

Splitting conjecture. There exist $R$ and $\mathbb{G}$ such that $\operatorname{Ext}(S^G;\Omega^1_{eq}(\hat{P};\hat{\mathfrak{g}}))$ is injective as an $R[\mathbb{G}]$-module, every $\operatorname{Ext}(S^G;\widehat{\operatorname{Conn}}(\hat{P};\hat{\mathfrak{g}}))$, except possibly the one with $\widehat{\operatorname{Conn}}(\hat{P};\hat{\mathfrak{g}})=\Omega^1_{eq}(\hat{P};\hat{\mathfrak{g}})$, is projective as an $R[\mathbb{G}]$-module, and the induced splittings can be chosen so that the corresponding map $s$ is continuous in the described topologies.

These algebraic hypotheses would make the restriction map fiberwise split and yield a continuous section embedding the extension bundle as an $R[\mathbb{G}]$-subbundle of a trivial bundle. The source does not establish the existence of the required $R$ and $\mathbb{G}$ or the continuous choices of splittings.