Isoperimetric inequality for spacelike star-shaped hypersurfaces in De Sitter space

Let $M\subset\mathbb S^{n+1}_1$ be a spacelike, compact, star-shaped, and $k$-convex hypersurface. For $1\leq k\leq n$, let $\mathcal A_k$ denote the quermassintegral-type quantity defined for such hypersurfaces, and let $\xi_{k,0}$ be the associated monotonically increasing function for radial coordinate slices. Isoperimetric conjecture. One has

$$\mathcal A_k\leq\xi_{k,0}(\mathcal A_0),\qquad 1\leq k\leq n,$$

with equality if and only if $M$ is a radial coordinate slice. The conjecture would follow from proving that the stated flow evolves every arbitrary $k$-convex hypersurface to a round sphere; it is therefore an isoperimetric inequality characterizing radial coordinate slices among the relevant hypersurfaces.