Primes of the power series rings

Question

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection \begin{equation*} \psi_{n,n-1} \colon A_n \twoheadrightarrow A_{n-1}. \end{equation*}

Choose a monic polynomial equation \begin{equation*} f(T) := T^e + a_{1}T^{e-1} + \ldots + a_{e-1}T + a_e, \end{equation*} where $a_1,\ldots,a_e \in (X_1,\ldots,X_n)A_n$. We shall consider the finite extension $B_n$ over $A_n$ defined as follows$\colon$

$B_n \colon= A_n[T]/(f(T))$.

By setting $B_{n-1} \colon= A_{n-1} \otimes_{A_n} B_n$, we have the following surjection induced by $\psi_{n,n-1}$$\colon$ \begin{equation*} \phi_{n,n-1} \colon B_n \twoheadrightarrow B_{n-1}. \end{equation*}

Suppose that a prime ideal ${\frak P}_n$ of $A_n$ satisfies the following surjection onto a prime ideal ${\frak P}_{n-1}$ of $A_{n-1}$$\colon$
\begin{equation*} (\sharp) \quad \psi_{n,n-1} \colon {\frak P}_n \twoheadrightarrow {\frak P}_{n-1}. \end{equation*}

Q. Suppose that ${\frak Q}_n$, ${\frak Q}_{n-1}$ are the unique prime ideals of $B_n$, $B_{n-1}$ lying over ${\frak P}_n$, ${\frak P}_{n-1}$, respectively. Moreover, suppose that both $B_n$, and $B_{n-1}$ are integral, i.e. domain. Then under the condition $(\sharp)$, does the following surjection still hold$\colon$

\begin{equation*} \phi_{n,n-1} \colon {\frak Q}_n \twoheadrightarrow {\frak Q}_{n-1}~ ? \end{equation*}