Unique geodesics and metric convexity for the space of compact spacelike Cauchy hypersurfaces

Let $(M,g)$ be a spatially compact spacetime, and let $\mathfrak{C}(M,g)$ denote its space of compact spacelike Cauchy hypersurfaces equipped with the $L^2$-metric. For $S_0,S_1\in\mathfrak{C}(M,g)$, consider geodesic paths in this Riemannian manifold joining them.

Geodesic uniqueness conjecture. Given two spacelike Cauchy hypersurfaces $S_0,S_1\in\mathfrak{C}(M,g)$, there is a unique geodesic path $c:[0,1]\to\mathfrak{C}(M,g)$ such that $c(0)=S_0$ and $c(1)=S_1$. Moreover, its length minimizes the lengths of paths from $S_0$ to $S_1$, and it is the only one to do so.

The conjecture is motivated by the non-positive sectional curvature of the $L^2$-metric, established in the paper, although existence and uniqueness of geodesics remain untreated because the geodesic equation is an infinite-dimensional partial differential equation with only weak ellipticity.