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?