Here $h_1, h_2, u_1, u_1, v_1, v_2$ are binary quadratic forms with integer coefficients, and $\Delta(h_1) = \Delta(h_2)$ (here $\Delta$ is the discriminant of a binary form). Put

$$\displaystyle F_1 = h_1(u_1(x,y), v_1(x,y)), F_2 = h_2(u_2(x,y), v_2(x,y)),$$

so that $F_1, F_2$ are binary quartic forms. Suppose that $F_1, F_2$ are $\operatorname{GL}_2(\mathbb{Z})$-equivalent; that is, there exists $T = \left(\begin{smallmatrix} t_1 & t_2 \\ t_3 & t_4 \end{smallmatrix}\right) \in \operatorname{GL}_2(\mathbb{Z})$ such that

$$\displaystyle F_1(t_1 x + t_2 y, t_3 x + t_4 y) = F_2(x,y),$$

and that $u_1, v_1, u_2, v_2$ are pairwise non-proportional. Does it follow that $h_1, h_2$ are $\operatorname{GL}_2(\mathbb{Z})$-equivalent?