Equality of critical thresholds for diffeomorphism groups

For a group of diffeomorphisms, define the distance threshold by

$$s^*_{\operatorname{dist}}=\sup\{s\in\mathbb R:d_s(g_1,g_2)=0\text{ for all }g_1,g_2\},$$

the smoothness threshold by

$$s^*_{\operatorname{smooth}}=\inf\{s\in\mathbb R:\operatorname{exp}\text{ is }C^1\},$$

and the Fredholm threshold by

$$s^*_{\operatorname{Fred}}=\inf\{s\in\mathbb R:v\mapsto\operatorname{ad}_v^{\star}u_0\text{ is compact}\}.$$

Here $d_s$ is the $H^s$ geodesic distance, $\operatorname{exp}$ is the Riemannian exponential map of the $H^s$ metric, and $v\mapsto\operatorname{ad}_v^{\star}u_0$ is the operator describing the Euler–Arnold equation. The critical-threshold conjecture. On any group of diffeomorphisms,

$$s^*_{\operatorname{smooth}}=s^*_{\operatorname{Fred}}\leq s^*_{\operatorname{dist}}.$$

The relation holds in all cases known to the authors, including diffeomorphism, volume-preserving, and symplectomorphism groups, but remains conjectural in general; behavior at the critical index can differ between properties.