The Lyapunov volume conjecture

For a compact Riemannian manifold $(N,g)$, define its Lyapunov volume by

$$\Lambda(N,g):=\sup_{(M,g|_M)\subset(N,g)\mid g|_M\text{ is non-trapping}}\operatorname{vol}_{g|_M}(M).$$

Lyapunov volume conjecture.

$$\Lambda(N,g)=\operatorname{vol}_g(N).$$

The claim says that the total volume can be approximated by non-trapping codimension-zero submanifolds; the source describes it as speculative and gives no resolution.