Conditions for being an entry in a trace compatible sequence

Question

$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal group law $F(X,Y)$ with coefficients in $\mathcal{O}_K$ for which $f$ is an endomorphism. Let $K_n$ be defined by adjoining the $n$-th level torsion points of $F$ to $K$ (i.e., the roots of $f^n(X)$). We say a sequence $(a_n)$ is trace compatible if $a_n \in K_n$ and $\Tr_{K_n/K_{n-1}}(a_n) = a_{n-1}$. If $\alpha \in K_n$ is arbitrary, what are some equivalent conditions to $\alpha$ being an entry in a trace compatible sequence? In particular can one say there exists a power of $\pi$, say $\pi^{k_\alpha}$, such that $\pi^{k_\alpha}\alpha$ is an entry in a trace compatible sequence?