Let $(G,X)$ be a Shimura datum of Hodge type. We say a point $x\in X$ is generic if the Mumford-Tate group of $x$ is the same as that of $X$ and special if the Mumford-Tate group of $x$ is a torus.
For a special point $x$, let $A_x$ be the corresponding abelian variety with CM and let $F$ be the field of definition. Let $v$ be a place of good reduction. Does there necessarily exist a generic point $y\in X$ such that the corresponding abelian variety $A_y$ has the same special fiber at $v$ as $A_x$?
If not, for what classes of Shimura varieties is this known?
