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.