Consider $\mathbb{C}$-algebras

$$A = \mathbb{C}[t][x_1,\ldots, x_n]\subset\mathbb{C}(t)[x_1,\ldots, x_n] = B$$

Group $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ carry a power series topology (see https://arxiv.org/abs/1712.01490 p.2).

Question. Fix $f\in B$. Is it true that $X =\{\pi\in\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])|\pi(f)\in A\}$ is closed in $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ with respect to power series topology?