Procongruence anabelian conjecture for congruence level structures

Let $\Bbbk$ be a sub-$p$-adic field with absolute Galois group $G_\Bbbk$. Let ${\mathcal M}^{\lambda}_\Bbbk$ and ${\mathcal M}^{\mu}_\Bbbk$ be moduli stacks of curves with congruence level structures, with geometric base points $\overline{\xi}$ and $\overline{\xi}'$. Let $\check{\pi}_1({\mathcal M}^{\lambda}_\Bbbk,\overline{\xi})$ and $\check{\pi}_1({\mathcal M}^{\mu}_\Bbbk,\overline{\xi}')$ be their procongruence fundamental groups, equipped with augmentation maps to $G_\Bbbk$, and let $\operatorname{Isom}_{G_\Bbbk}(-,-)^\mathrm{out}$ denote compatible isomorphisms modulo the inner action of the geometric procongruence fundamental group. Procongruence anabelian conjecture for congruence level structures. There is a natural isomorphism of torsors

$$\operatorname{Isom}_\Bbbk({\mathcal M}^{\lambda}_\Bbbk,{\mathcal M}^{\mu}_\Bbbk)\stackrel{\sim}{\to}\operatorname{Isom}_{G_\Bbbk}(\check{\pi}_1({\mathcal M}^{\lambda}_\Bbbk,\overline{\xi}),\check{\pi}_1({\mathcal M}^{\mu}_\Bbbk,\overline{\xi}'))^\mathrm{out}.$$

This is the procongruence analogue of the étale anabelian conjecture, replacing the full étale fundamental groups by their congruence completions; the supplied text gives no resolution status.