Orthogonal matrix decomposition

Question

Consider an orthogonal matrix $\boldsymbol{A}$ that can be decomposed as $\boldsymbol{A}=\boldsymbol{BCD}$, where $\boldsymbol{A}$, $\boldsymbol{B}$, $\boldsymbol{C}$, $\boldsymbol{D} \in \mathbb{C}^{N\times N}$. Suppose $\boldsymbol{C}$ is a diagonal matrix with normalized complex exponential diagonal elements, and all the elements of $\boldsymbol{B}$ and $\boldsymbol{D}$ share a common modulus. Then must $\boldsymbol{B}$ and $\boldsymbol{D}$ be scaled orthogonal matrices?