Mostow rigidity for complex hyperbolic manifolds A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.

Theorem (Real Mostow Rigidity) If $X$ and $Y$ are closed, hyperbolic $n$-manifolds with $n \ge 3$ and $\pi_1 X \simeq \pi_1 Y$ then there is a Riemannian homothety $\varphi:X \simeq Y$.

A Riemannian homothety is a diffeomorphism $\varphi:X \simeq Y$ with $\varphi^*g_Y = \lambda \cdot g_X$ for some $\lambda > 0$. 
There is a clear analogue of the hypothesis of Mostow rigidity in the Kahler setting. Namely, a Kahler manifold $(X,\omega,J)$ is complex-hyperbolic if the holomorphic sectional curvatures are constant and negative. For a definition of the holomorphic sectional curvatures, see (for instance) Hyperbolic Complex Spaces by Kobayashi.
Main Question: My question is whether or not the following Kahler analogue is true.

Theorem (?) (Complex Mostow Rigidity) If $X$ and $Y$ are closed, complex-hyperbolic $2n$-manifolds with $\pi_1 X \simeq \pi_1 Y$ then there is a Kahler homothety $\varphi:X \simeq Y$.

A dimensional restriction may be necessary here, for instance $n \ge 2$ or $n \ge 3$. Again, I would like to know if this theorem is true and if so, where I can find a reference.
 A: The general statement of Mostow-Prasad rigidity cited from http://repository.ias.ac.in/36364/1/36364.pdf is as follows.
Let $G$ (resp. $G^\prime$) be a semi-simple analytic group and $\Gamma$ (resp. $\Gamma^\prime$) an irreducible lattice in $G$ (resp. $G^\prime$). Assume that $G, G^\prime$ have trivial centers and no compact factors and $G$ is not locally isomorphic to $SL(2,{\mathbb R})$. Then any isomorphism $\theta\colon \Gamma\to\Gamma^\prime$ extends to an analytic isomorphism of $G$ onto $G^\prime$.
The case of complex hyperbolic manifolds corresponds to $G=G^\prime=SU(n-1,1)$. An analytic isomorphism is an isometry and Kähler.
A: If $X$ is closed, complex hyperbolic and $Y$ is only assumed to be Kähler, aspherical and with the same fundamental group, then $X$ and $Y$ have the same homotopy type (they are $K(G,1)$'s for the same $G$).
A theorem of Siu (Annals of Math. 1980) then gives that $X$ and $Y$ are biholomorphic (or anti-biholomorphic).
(If $Y$ is complex hyperbolic this map will turn out to be a homothety.)
Siu's theorem can be thought of as a strengthening of Mostow's theorem, in the Kähler category. It can also be stated with $X$ being the quotient of any Hermitian symmetric space of noncompact type.
