Take some stable theory $T$ with elimination of imaginaries, all sets appearing are small subsets of the monster model of $T$, elementary maps are restrictions of automorphism of the monster model of $T$.

Is the following statement correct?

For all algebraic closed sets $A,B$, with additionaly that these sets are independent over $acl(\emptyset)$. And all elementary $f:acl(AB)\rightarrow acl(AB)$ fixing $A$ and $B$. And all $C,D$ two (closed) sets independent over $acl(\emptyset)$ with $A\subset C$ and $B\subset D$. We can extend $f$ to some automorphism fixing $C$ and $D$.

Or equivalently, there is no $c\in acl(A\cup B)-dcl(A\cup B)$, such that $Ac$ independent of $B$ over $acl( \emptyset)$ and $A$ not independent of $Bc$ over $acl(\emptyset)$ (or vice versa).