Complete distance characterization for volume-preserving diffeomorphisms

Let $M$ be a manifold of dimension at least $3$, and let $\operatorname{Diff}_\mu(M)$ denote its group of volume-preserving diffeomorphisms equipped with the right-invariant $H^s$ metric. The volume-preserving distance conjecture. The geodesic distance vanishes if and only if

$$s<0.$$

This is presented as a consequence that would follow if the critical-threshold conjecture were true, together with known Fredholmness results; it is therefore conditional and remains open in the stated generality.