The solvable Homeomorphism Conjecture for moduli spaces of fundamental groups

Let $p>0$ be the characteristic. Retain the Frobenius-quotiented moduli space $\overline{\mathfrak{M}}_{g,n}$ of pointed stable curves of type $(g,n)$ and let $\overline\Pi^{\rm sol}_{g,n}$ be the moduli space of solvable admissible fundamental groups of pointed stable curves of type $(g,n)$, equipped with its anabelian topology. Let

$$\pi_{g,n}^{\rm sol}:\overline{\mathfrak{M}}_{g,n}\twoheadrightarrow\overline\Pi^{\rm sol}_{g,n}$$

be the natural map. Solvable Homeomorphism Conjecture. The map $\pi_{g,n}^{\rm sol}$ is a homeomorphism. This is presented as a slightly stronger solvable version of the Homeomorphism Conjecture. The corresponding set-theoretic bijection is likewise formulated as a solvable version of the Weak Isom-version Conjecture, but the homeomorphism assertion remains open in the supplied text.