Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions related by 
$$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \qquad \forall z\in D.$$

>**Question.** If $f\neq \alpha g$ for any $\alpha\in\mathbb{C}$ then is the same true for $F$ and $G$?