Narkiewicz's conjecture on property $(P)$ for $[200~\mathbb{Q}^{(d)}$[201~

For a field $F$, define property $(P)$ by requiring that every polynomial $g(x)\in F[x]\setminus F$ for which there is an infinite set $X\subseteq F$ satisfying $g(X)=X$ has degree one. For a positive integer $d$, let $F^{(d)}$ be the compositum in an algebraic closure of all field extensions of $F$ of degree at most $d$. Narkiewicz's conjecture. The field $\mathbb{Q}^{(d)}$ has property $(P)$ for all positive integers $d$. This is the Narkiewicz conjecture discussed in the paper; the surrounding text states that it is still open.