Weighted horospherical p-Minkowski inequalities

Let $K,L\subset\mathbb H^{n+1}$ be smooth origin-symmetric, uniformly h-convex bounded domains, with $n\geq1$. Let $\varphi_K,\varphi_L$ be their horospherical support functions, let $r$ be the hyperbolic distance from the origin, and let $\tilde u$ denote the support quantity appearing in the weighted integrand. Define $\mathcal S$ as the normalized weighted volume functional. Weighted horospherical $p$-Minkowski conjecture. For $p>0$,

$$\int_{\partial K}\varphi_L^p\cosh r(\cosh r-\tilde u)^p\,d\mu-\int_{\partial K}\cosh r\,d\mu\\ \geq \omega_n\mathcal S^n(K)\sqrt{\mathcal S^2(K)+1}\left[\left(\frac{\mathcal S(L)+\sqrt{\mathcal S^2(L)+1}}{\mathcal S(K)+\sqrt{\mathcal S^2(K)+1}}\right)^p-1\right],$$

and for $p<0$ the inequality is reversed. In both cases equality should hold exactly when $K=L$ or $K,L$ are geodesic balls centered at the origin. This is the weighted infinitesimal counterpart of the weighted horospherical Brunn–Minkowski conjecture.