Quermassintegral inequalities conjecture for spacelike domains in de Sitter space

Let $K$ be a smooth bounded domain with $(k-1)$-convex and spacelike boundary $\partial K$ in $\mathbb S_{+}^{n,1}$. Let $W_k(K)$ denote its $k$-th quermassintegral, and let $f_k$ and $f_l$ be the functions appearing in the inequality.

Quermassintegral inequalities conjecture. There hold

$$W_k(K)\leq f_k\circ f_l^{-1}(W_l(K)),\qquad 0\leq l<k\leq n.$$

Equality holds if and only if $\partial K$ is isometric to a coordinate slice. These inequalities are proposed as the de Sitter-space analogues of quermassintegral inequalities in hyperbolic space; the statement and its equality characterization remain open in the supplied source.