On a symplectic manifold $(M,\omega)$, is it true for any $2$-form $\mu$, that $\mu=0$ iff $\omega^{n-1}\wedge\mu=0$?

Question

In this paper, on page 8 (above Theorem 9), I found the following claim::

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold. If $\mu$ is any $2$-form on $M$, then $$\omega^{n-1}\wedge\mu=0\Leftrightarrow \mu=0.$$

The implication $\Leftarrow$ is clear, but I have trouble understanding why the other implication holds. Could somebody shine some light on this?

Answer 1

It is a mistake in the paper, as the counterexamples in the comments section demonstrate.