Weighted quermassintegral conjecture for static convex domains

Let $\Omega$ be a static convex domain with smooth boundary in $\mathbb H^{n+1}$. For $0\leq l<k\leq n+1$, define

$$h_k(r)=W_k^{\lambda'}(B_r)=\omega_n\sinh^{n+1-k}r\cosh^k r,$$

where $B_r$ is the geodesic ball of radius $r$, and let $h_l^{-1}$ denote the inverse of the monotone function $h_l$. The weighted curvature integrals are

$$W_{0}^{\lambda'}(\Omega)=\int_M u\,d\mu,\qquad W_k^{\lambda'}(\Omega)=\int_M\lambda' E_{k-1}\,d\mu\quad(1\leq k\leq n+1).$$

Weighted quermassintegral conjecture. One has

$$W_k^{\lambda'}(\Omega)\geq h_k\circ h_l^{-1}\bigl(W_l^{\lambda'}(\Omega)\bigr).$$

Equality holds if and only if $\Omega$ is a geodesic ball centered at the origin. This would extend the known weighted isoperimetric and Alexandrov–Fenchel inequalities for static convex domains in hyperbolic space to all pairs $0\leq l<k\leq n+1$.