A weighted quermassintegral inequality involving weighted quermassintegrals for strictly convex domains in the sphere

Let $\Omega$ be a strictly convex domain with smooth boundary $M$ in $\mathbb{S}^{n+1}$. For integers $k$ and $l$ satisfying $0\leqslant k\leqslant n$ and $0\leqslant l\leqslant k$, let $\Phi$, $p_k$, $W_{k-1}$, $\xi_k$, $f_{k-1}$, $h_l$, and $W_l^{\phi'}$ denote the quantities appearing in the inequality. The weighted quermassintegral conjecture. One has

$$\int_{M}\Phi p_kd\mu+kW_{k-1}(\Omega)\geqslant (\xi_k+kf_{k-1})\circ h_l^{-1}(W_l^{\phi'}(\Omega)).$$

Equality holds if and only if $\Omega$ is a geodesic ball centered at the origin. The conjecture is one of three questions raised after the preceding results; the supplied text gives no resolution or further context.