Associated-connection conjecture for Lie group fiber bundles

Let $(\mathcal{G}, M, \pi_{\mathcal{G}, M}, G)$ be a Lie group fiber bundle with connected typical fiber $G$, and let $\eta$ be a Lie group fiber bundle connection on it. Let $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$ be the principal bundle constructed in the proof of Proposition 1.3.4.

Associated-connection conjecture. The connection $\eta$ can be seen as an associated connection to a principal connection on $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$.

This would identify Lie group fiber bundle connections with connections induced from principal connections on the canonical principal bundle associated with the bundle. The supplied text presents this as a natural question, but gives no resolution.