Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space. 
This space is a complex analog of hyperbolic space.  It is isometric to the quotient of hyperboloid
$$|z_0|^2-|z_1|^2-\dots-|z_n|^2=1$$
in $\mathbb{C}^{n+1}$
by $S^1$.

>**Question 1.** Is it known that round balls in $\mathbb{H}_\mathbb{C}^n$ minimize the surface area among all bodies of given volume? 

(I am almost sure that the answer is not known.)

>**Question 2.** Was it conjectured somewhere?