Brendle–Guan–Li Alexandrov–Fenchel conjecture for k-convex star-shaped hypersurfaces in the sphere

Let $M$ be a smooth, closed, connected, embedded, $k$-convex, star-shaped hypersurface in the sphere $\mathbb{S}^{n+1}$ enclosing a bounded domain $\Omega$. Let $\mathcal{A}_k(\Omega)$ denote the corresponding quermassintegral, and let $s_{k-1}$ denote the relevant upper endpoint. Brendle–Guan–Li's Alexandrov–Fenchel conjecture. One has

$$\mathcal{A}_k(\Omega)\geq \xi_{k,k-1}\big(\mathcal{A}_{k-1}(\Omega)\big),$$

where $\xi_{k,k-1}$ is a unique positive function defined on $(0,s_{k-1})$, with equality for geodesic spheres; equality holds if and only if $M$ is a geodesic sphere. Alexandrov–Fenchel inequalities for hypersurfaces in the sphere have long been open. The conjecture was proposed by Brendle, Guan, and Li; only the convex case $k=n-1$ was previously confirmed, while the general $k$-convex star-shaped case remains unresolved in the supplied text.