A more general version of the Fejér-Riesz theorem

Question

A classical result, known as the Fejér-Riesz theorem, states that any Laurent polynomial $p(z)=\sum_{|k|\leq N} c_kz^k$ (the coefficients $c_k$ are complex numbers) which is nonnegative on the torus $\mathbb{T}:=\{z\in\mathbb{C}: |z|=1\}$, namely $p(z)\geq 0$ for every $z\in\mathbb{T}$, can be written as $$p(z)=|q(z)|^2$$ where $q$ is still a Laurent polynomial. Now my question is whether a more general version of this theorem may continue to hold when the coefficients are allowed to be continuous functions on some compact space $X$. More precisely, we now consider a complex function $p$ defined on the product $X\times\mathbb{T}$ of the type $$p(x, z)=\sum_{|k|\leq N}c_k(x)z^k$$ where each $c_k$ sits in $C(X)$, the (complex-valued) continuous function on $X$. If $p(x, z)\geq 0$ for every $(x, z)\in X\times\mathbb{T}$, is it still true that there exists a function $q$ of the same type as $p$ such that $p(x, z)=|q(x, z)|^2$ for every $(x, z)\in X\times\mathbb{T}$? My guess is the answer might possibly depend on the space $X$, but I haven't been able to find anything helpful (for instance, the so-called operator Fejér-Riesz theorem doesn't seem to apply to this situation).