Let $f,g \in \mathbf {C}[x_1, \ldots, x_n]\subseteq \mathbf {C}[x_1, \ldots, x_{n+1}]$ be two polynomials with complex coefficients and suppose that there exists $h_1 \in \mathop{GL}_{n+1}(\mathbf C)$ such that $h_1 \cdot f=g$. Does there exist $h_2 \in \mathop{GL}_n(\mathbf C)$ such that $h_2\cdot f=g$?

This is a toy example of a more general question I'm interested in: given a family of representations $\rho_n: \mathop{GL}_n \to \mathop{End}(V_n)$ such that $V_n$ embeds naturally in $V_{n+1}$ (e.g., $V_n=\Lambda^r(\mathbf C^n)$), what can be said about the $\mathop{GL}_{n+1}$-orbits of elements of $V_n$?