Non-local exponentiality conjecture for semidirect products of smooth functions

Let $M$ be a compact manifold and let

$$\sigma \colon {\mathbb{R}} \times M\to M$$

be a smooth non-trivial flow. Define an action of ${\mathbb{R}}$ on $C^\infty(M,{\mathbb{R}})$ by

$$\alpha_t(f) \colonequals f \circ \sigma_t.$$

Non-local exponentiality conjecture. The Lie group $C^\infty(M,{\mathbb{R}}) \rtimes_\alpha {\mathbb{R}}$ is not locally exponential. More specifically, there is a sequence $t_n \in {\mathbb{R}}_+$ with $t_n \to 0$ such that the points $(0,t_n)$ are singular for the exponential function.

The preceding results establish non-local exponentiality under several conditions on the flow, but do not completely characterize it for arbitrary smooth flows on compact manifolds. This conjecture proposes that every non-trivial flow has singularities of the exponential function arbitrarily close to the identity.