Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field. Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$. Assume that for every $k$-point of $A$, the algebraic group $G_k$ has a maximal $r$-dimensional $k$-torus.

Question.Does it follow that $G$ has a maximal $r$-dimensional $A^\times$-torus?