Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]

Question

Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation

$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = 0$ $\phantom{a}$ with $\phantom{a}$ $a_1,\ldots,a_n \in A_3$.

We shall denote by $S(\sharp)$ the set of whole solutions of $(\sharp)$ such that $Y_1,\ldots,Y_n \in A_3$.

Definition. We shall define ``specialisation'' as substituting $X_3$ with some element $F(X_1,X_2) \in K[[X_1,X_2]]$.

After specialisation with $F(X_1,X_2)$, we obtain a linear equation

$(\sharp)_F \phantom{aa} a_1(X_1,X_2,F(X_1,X_2))Y_1 + \ldots + a_{n}(X_1,X_2,F(X_1,X_2))Y_n = 0$.

We shall denote by $S((\sharp)_F)$ the set of whole solutions of $(\sharp)_F$ such that $Y_1,\ldots,Y_n \in K[[X_1,X_2]]$.

By inserting $F(X_1,X_2)$ for $X_3$, we obtain a natural map $S(\sharp) \to S((\sharp)_F)$.

Q. For a given linear equation $(\sharp)$, does there always exist some specialisation $X_3 = F(X_1,X_2)$ such that the natural map $S(\sharp) \to S((\sharp)_F)$ is a surjection?