A hyperbolic Michael–Simon inequality for higher mean curvatures

Let $M$ be a compact hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. For $1\le k\le n$, let $E_k=E_k(\kappa)$ denote the normalized $k$-th mean curvature and let $E_{k-1}$ be defined similarly; write $\lambda'$ for the radial derivative factor, $\nabla^M f$ for the intrinsic gradient, $\nu$ for the unit outward normal, $u$ for the support function, and $\bar{\nabla}$ for the Levi-Civita connection of the hyperbolic metric. Then there is a constant $C$ such that

Higher-mean-curvature hyperbolic Michael–Simon conjecture. For $1\le k\le n$, the inequality

$$\int_M\lambda'\sqrt{f^2E_k^2+|\nabla^M f|^2E_{k-1}^2}-\int_M\left\langle\bar\nabla(f\lambda'),\nu\right\rangle|E_{k-1}|+\int_{\partial M}f|E_{k-1}| \ge C\left(\int_M f^{\frac{n-k+1}{n-k}}|E_{k-1}|\right)^{\frac{n-k}{n-k+1}}$$

should hold. In particular, when $M$ is closed and $f$ is constant,

$$\int_M(\lambda'|E_k|-u|E_{k-1}|)\ge C\left(\int_M|E_{k-1}|\right)^{\frac{n-k+1}{n-k}}.$$

This is proposed as the higher-mean-curvature analogue of the preceding hyperbolic Michael–Simon inequality. The source does not provide a proof or a resolution, and the constant $C$ is not further specified.