$P(P(a)) = a$. Are $a$ and $P(a)$ Galois conjugates?

Question

Let $P \in \mathbf{Z}[x]$ monic and $V(x) = \underbrace{P \circ \ldots \circ P}_n(x)-x$.

If $a$ is a root of $V$, are $a$ and $P(a)$ Galois conjugates ?

Edit : user608470 found a counterexample. Do you think there is a chance to understand where the counterexamples come from ?

I found no counterexample with this magma script

R<x> := PolynomialRing(Integers());   P := x^4+x^3+1; n := 4;
Pn := x; for m in [1..n] do    Pn := Evaluate(P,Pn); end for; V := Pn-x;
f := Factorization(V); 
for l in [1..#f] do
  h := f[l][1]; // a's minimal polynomial
  (Evaluate(h,P) mod h) eq 0; // true so P(a) is a root of a's minimal polynomial
end for;

If $a,P(a)$ are conjugates in a finite field $\mathbf{Z}[a]/ \mathfrak{P}$ and $V'(a) \not \equiv 0 \bmod \mathfrak{P}$ then by Hensel lemma $a,P(a)$ are conjugates in the $p$-adic field $\varprojlim \mathbf{Z}[a]/\mathfrak{P}^m$ so they are also Galois conjugates over $\mathbf{Z}$.

Note $2 \in \mathbf{F}_5$ is a root of $P(P(P(x)) -x \in \mathbf{F}_5[x]$ with $P(x) = x^2+1$ and it isn't conjugate to $2^2+1$ so if the answer to the question is yes, it depends on special properties of $\mathbf{Z}$.