Alexandrov-Fenchel inequality for odd curvature integrals

Let $n\geq 3$, let $k$ be an integer satisfying $2k+1\leq n-1$, and let $\Sigma$ be a hypersurface in hyperbolic space $\mathbb{H}^n$ with nonnegative sectional curvature. Write $p_j$ for the normalized $j$-th elementary symmetric curvature of $\Sigma$, $|\Sigma|$ for its area, and let $\omega_{n-1}$ denote the area of the unit $(n-1)$-sphere.

Alexandrov-Fenchel inequality. One should have

$$\int_{\Sigma} p_{2k+1}\geq \omega_{n-1}\left[\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{2}{2k+1}}+\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{2}{2k+1}\frac{n-2-2k}{n-1}}\right]^{\frac{2k+1}{2}}.$$

Equality should hold if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$.

This proposes an Alexandrov-Fenchel-type inequality under nonnegative sectional curvature, extending the preceding inequality for $\int_{\Sigma}p_1$. The paper presents it as an open conjecture; whether the relevant curvature conditions are preserved along inverse mean curvature flow remains an open problem.