Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.

Consider

$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$

the first Chern class of a line bundle (note that the Picard coincides with $H^2(\overline{\mathcal{M}}_{g,n}, \mathbb{Z})$ and with the Neron-Severi group in this case)

It is well-known from Arbarello-Cornalba-Harer that the group of first Chern classes is generated by $\kappa_1$ and by $\delta$-classes and $\psi$-classes.

My question: is the multiplication by $\gamma$ an isomorphism:

$ H^{3g-3+n-1} \to H^{3g-3+n+1} \ ? $

(as it happens in the Hard Lefschetz theorem when $\gamma$ is an hyperplane section). Are there $\gamma$s for which it is not an isomorphism?

This is of course true when $\gamma$ is in the ample cone or in the antiample cone.