Fejer-Riesz Theorem and applications

Question

Let $\mathbb{D}=\{z: |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose $D$ and $E$ are polynomials of degree atmost $n$ with complex coefficients such that

1. $|E(z)| \leq |D(z)|$ for all $z \in \overline{\mathbb{D}}$
2. $D(z) \ne 0$ for all $z \in \mathbb{D}$.
3. $E(z)=E^+(z) \quad \text{where} \quad E^+(z)=z^n\overline{E(1/\overline{z})}$ for all $z \in \mathbb{C}$.
4. $z \mapsto |E(z)|^2-|D(z)|^2$ for $z \in \mathbb{T}$ has only finitely many zeros
5. $E(0)=0$ and $D^+(z)=0 \quad \text{where} \quad D^+(z)=z^n\overline{D(1/\overline{z})}$
6. $|D(0)|=1$.

Prove that there exists a polynomial $A$ that does not vanish on $\mathbb{D}$ and $|A(z)|^2=|D(z)|^2-|E(z)|^2$ for all $z \in \overline{\mathbb{D}}$.

My attempt: First note that for $z \in \mathbb{T}$, the function $Q(z)=|E(z)|^2-|D(z)|^2$ is a trigonometric polynomial. By given hypothesis, $Q(z) \geq 0$ on $\mathbb{T}$ and $Q$ is not identically zero. By Fejer-Riesz Theorem, there exists a polynomial $A$ that does not vanish on $\mathbb{D}$ and $|A(z)|^2=|D(z)|^2-|E(z)|^2$ for all $z \in \mathbb{T}$. Let us consider the polynomial map

$R(z)=A(z)A^+(z)-D(z)D^+(z)+E(z)E^+(z)$.

Then, for every $t \in \mathbb{R}$,

$|R(e^{it})|=|A(e^{it})|^2-|D(e^{it})|^2 + |E(e^{it})|^2=0$.

Hence, by identity theorem, $R(z)=0$ for all $z \in \mathbb{C}$.

If I am able to prove $|A(0)|=1$, then $|A(0)|^2=|D(0)|^2-|E(0)|^2$. From here, I can use some maximum principle in higher variable to conclude the desired result.

How to proceed further? Any hints/ suggestions.