Do there exist regular symmetric differential forms on $\overline{\mathcal{M}}_{g,n}$ the DM-stack of stable genus $g$ curves with $n$ marked points? By this, I mean nonzero sections $$ \omega \in H^0(\overline{\mathcal{M}}_{g,n}, \mathrm{Sym}^k \Omega^1_{\overline{\mathcal{M}}_{g,n}}) $$

It is shown here (or by vanishing results fro $H^1(\overline{\mathcal{M}}_{g,n})$) that there are no such forms in the case $k = 1$. I am wondering about the case $k > 1$.