Regularity conjecture for the stable moduli hypersurfaces

Let $\mathfrak M$ be a Banach-space moduli space of initial data, and for $\mathfrak r\in[0,1]$ let $\mathfrak M^\mathfrak r_\mathrm{stab}\subset\mathfrak M$ consist of initial data forming a black hole with asymptotic parameter ratio $|e_\infty|/M_\infty=\mathfrak r$. Let $\mathfrak M_\mathrm{black}$ be the subset of seed data forming a black hole in evolution. Regularity conjecture. For every $\mathfrak r\in[0,1]$, $\mathfrak M^\mathfrak r_\mathrm{stab}$ is a $C^1$ hypersurface in $\mathfrak M$. Moreover, $\mathfrak M_\mathrm{black}$ is foliated by the collection $\{\mathfrak M^\mathfrak r_\mathrm{stab}\}$. This conjectural picture concerns the local structure of the moduli space; the stable set associated with extremal black holes is expected in particular to be a connected regular hypersurface, but the asserted regularity and foliation are not established here.