Horospherical p-Brunn–Minkowski inequalities for modified quermassintegrals

Let $n\geq 1$ and $0\leq k\leq n$ be integers, let $\frac12\leq p\leq 2$, and let $a,b\geq 0$ satisfy $a+b\geq 1$. For smooth uniformly h-convex bounded domains $K,L\subset\mathbb{H}^{n+1}$, set $\Omega=a\cdot K+_p b\cdot L$. For each h-convex bounded domain $E$, define $r^k_E=I_k^{-1}(\widetilde{W}_k(E))$. Horospherical $p$-Brunn–Minkowski conjecture.

$$\exp\left(pI_k^{-1}(\widetilde{W}_k(\Omega))\right)\geq a\exp\left(pI_k^{-1}(\widetilde{W}_k(K))\right)+b\exp\left(pI_k^{-1}(\widetilde{W}_k(L))\right),\qquad k=0,1,\ldots,n.$$

For $0\leq k\leq n-1$, equality should hold exactly in the four listed cases: $a=1,b=0$; $a=0,b=1$; $K=L$ and $a+b=1$; or $K,L$ are geodesic balls centered at the same point. For $k=n$, equality should hold exactly when $a\geq1,b=0$; $a=0,b\geq1$; or $K,L$ are hyperbolic dilates. The conjecture extends the proved inequality for geodesic balls to general smooth uniformly h-convex domains.