Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow \overline{\mathcal{M}}_{0,n}$. Is it a submersion away from the nodal points of its fibres?
I've tried working with the description of $\overline{\mathcal{M}}_{0,n}$ as a subset of a product of spheres but the description of the tangent spaces is not very accessible.
