The cactus flower moduli-space homeomorphism conjecture

Let $\breve D_n$ be the cube complex whose cubes are indexed by $[n]$-labelled cyclic forests, and let $\overline M_{n+2}^\sigma(\mathbb{R})$ be the indicated real locus of the moduli space $\overline M_{n+2}$. Its geometric strata are indexed by the same cyclic forests, with codimension equal to the number of internal edges. Cactus flower homeomorphism conjecture. There is a homeomorphism

$$\breve D_n \cong \overline M_{n+2}^\sigma(\mathbb{R})$$

such that the cube complex and the geometric stratification are dual complexes. The conjecture identifies the combinatorial compactification encoded by $\breve D_n$ with the real moduli space and its stratification; the supplied text gives no resolution status.