Let $(K,\sigma)$ be a difference field, ie equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $\delta=0$ for some $\delta\in K\{x\}$, where $$K\{x\}=K\left[\partial^i(x):i\in\mathbb N\right].$$ Let $F=\{x\in K:\sigma x=x\}$ be the subfield fixed by $\sigma$.

Question. Is $[K:F]$ finite?

Remarks. (1) The answer is yes if the identity is of 'degree 1', i.e. of the form $$\sum_{k=0}^n a_k\sigma^k(x)=a_{{-1}}$$ for some $a_{-1},a_0,\dots,a_n\in K$, and conversely, a finite dimensional $F$-algebra satisfies such an identity.

(2) It is also yes if $char K=p>0$ and $\sigma$ is the Froebenius...

Edit. The following question is motivated by the question:

Question. Is the evaluation map that maps a difference polynomial $\delta\in K\{x\}$ to the function $\delta_K:K\rightarrow K$ is injective, at least when $[K:F]$ is infinite?