This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.

By Mostow Rigidity Theorem, the geometry of a hyperbolic manifold $X$ of real dimension at least $3$ is determined by its fundamental group, and hence it is unique. As a corollary, it follows that a complex ball quotient $X$ of (complex) dimension at least $2$ (for instance, a fake projective plane) has no deformations in the large (i.e. is *rigid*), in other words the base $\textrm{Def}(X)$ of the Kuranishi family of $X$ consists of a single point.

Question.Is it also true that $X$ isinfinitesimally rigid, i.e. that $H^1(X, \, T_X)=0$?

If $H^1(X, \, T_X) \neq 0$ then by Mostow's theorem it would follow that $X$ has *obstructed* first order deformations. I guess this is not the case, but I was unable to find a precise reference or statement in the wealth of results concerning deformations (or rigidity) of hyperbolic manifolds.