Moduli spaces of witch curves topologically realize the 2-associahedra

For $r\ge 1$ and $$\mathbf{n}$ \in $\mathbb{Z}${\geq0}^r\setminus{$\mathbf{0}$}$, we construct the compactified moduli space ${\overline{2\mathcal{M}}}_{\mathbf{n}}$ of witch curves of type $\mathbf{n}$. We equip ${\overline{2\mathcal{M}}}_{\mathbf{n}}$ with a stratification by the 2-associahedron $W_{\mathbf{n}}$, and prove that ${\overline{2\mathcal{M}}}_{\mathbf{n}}$ is compact and metrizable. In addition, we show that the forgetful map $\overline{2$\mathcal{M}$}{$\mathbf{n}$} \to \overline{$\mathcal{M}$}_r$ to the moduli space of stable disk trees is continuous and respects the stratifications.