Weighted Alexandrov–Fenchel conjecture for star-shaped $k$-convex domains

Let $0\leq k\leq n$ and $0\leq m\leq k$. Let $\Omega$ be a star-shaped and $k$-convex domain in $\mathbb H^{n+1}$ with smooth boundary $M$. Let $W_{k+1}^{\lambda'}(\Omega)=\int_M\lambda'E_k\,d\mu$, and let $h_{k+1}$ and $f_m$ be the monotone geodesic-ball comparison functions used for the weighted and ordinary curvature integrals, respectively. Weighted Alexandrov–Fenchel conjecture. One has

$$W_{k+1}^{\lambda'}(\Omega)=\int_M\lambda'E_k\,d\mu\geq h_{k+1}\circ f_m^{-1}\bigl(W_m(\Omega)\bigr).$$

Equality holds if and only if $\Omega$ is a geodesic ball centered at the origin. The corresponding inequality is proved in the paper for static convex domains; the conjecture asks for the star-shaped, $k$-convex case.