Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety of $Y$ such that for any two points of $Y$ the corresponding fibers are biholomorphic.

Question. Is it true that for any two general points of $Y$ the fibers are isomorphic ?

In other words, are the leaves of the foliation defined by the image of $\pi_* TX$ in $TY$ algebraic ?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two points of $Z$ the corresponding fibers are biholomorphic.

Question. Is it true that for any two general points of $Y$ the corresponding fibers are biholomorphic ?

In other words, are the leaves of the foliation defined by the image of $\pi_* TX$ in $TY$ algebraic ?