Horospherical p-Minkowski inequalities of types I and II

Let $n\geq1$, let $0\leq k\leq n$, and let $K,L\subset\mathbb{H}^{n+1}$ be smooth uniformly h-convex bounded domains. Write $\varphi_K,\varphi_L$ for their horospherical support functions, $\tilde\lambda_K$ for the shifted principal-curvature data, and $r^k_E=I_k^{-1}(\widetilde W_k(E))$. Horospherical $p$-Minkowski conjecture. For real $p$, the source proposes the displayed inequalities and equality cases for $p>0$ and for $-n\leq p<0$:

$$\int_{\mathbb S^n}\varphi_L^p\varphi_K^{-p-n}p_{n-k}(\tilde\lambda_K)\,d\sigma-\int_{\mathbb S^n}\varphi_K^{-n}p_{n-k}(\tilde\lambda_K)\,d\sigma\geq\omega_n\sinh^{n-k}r^k_K e^{-kr^k_K}\left(e^{p(r^k_L-r^k_K)}-1\right)$$

when $p>0$, with the reverse inequality when $-n\leq p<0$. The stated equality cases are exactly those in the source: for $0\leq k\leq n-1$, $K=L$ or concentric geodesic balls in the positive and $-n<p<0$ cases; for $p=-n$ the negative case allows geodesic balls; for $k=n$ they are the stated dilate cases, with equality always when $k=n,p=-n$. These conjectures are infinitesimal forms of the horospherical Brunn–Minkowski conjecture; the $k=n$ Brunn–Minkowski case has already been solved, while the remaining cases are open.