Bauer's geodesic-completeness conjecture for uniformly stronger weak Riemannian metrics

Let $G^1$ and $G^2$ be smooth, weak Riemannian metrics on an infinte-dimensional Banach manifold $\mathcal H$. Assume that $G^2$ is uniformly stronger than $G^1$, both metrics admit a smooth geodesic spray and hence have locally diffeomorphic Riemannian exponential maps, and $G^1$ is geodesically complete, meaning that every geodesic initial value problem has solutions for all time $t$. Possible additional assumptions are that $\mathcal H$ is a topological group with both metrics right invariant, and that derivatives of $G^1$ are controlled by derivatives of $G^2$. Bauer's geodesic-completeness conjecture. Under these assumptions, $G^2$ is geodesically complete. The conjecture concerns completeness of weak Riemannian metrics on infinite-dimensional manifolds; the source presents the listed conditions as minimal assumptions that might need supplementation, so its general validity remains open.