Collapse of the Fω Typability Hierarchy

For $n\geq 0$, let $F_n$ be the fragment of System $F_\omega$ in which type constructors have order at most $n$; thus $F_0$ is System $F$. For every pure lambda term $M$, does

$$M\text{ typable in }F_\omega \quad\Longrightarrow\quad M\text{ typable in }F_1$$

hold?