Anderson's generic regularity conjecture for the boundary map of Einstein 3-manifolds

Let $\Omega$ be a $3$-dimensional manifold with smooth boundary $\Sigma$ satisfying $\pi_1(\Omega,\Sigma)=0$. For fixed $k$ and $\alpha$, let $\mathcal{M}^{k,\alpha}_\Lambda(\Omega)$ denote the moduli space of Einstein metrics with cosmological constant $\Lambda$, modulo the group $\mathscr D^{k+1,\alpha}(\Omega)$ of $\mathcal{C}^{k+1,\alpha}$ diffeomorphisms restricting to the identity on $\Sigma$. For a metric $g$, write $g^\intercal$ for its induced boundary metric, let $[g^\intercal]$ be its conformal class, let $H_g$ be its boundary mean curvature, and let $\mathcal{S}_1^{k,\alpha}(\Sigma)$ be the space of conformal classes of $\mathcal{C}^{k,\alpha}$ Riemannian metrics on $\Sigma$, identified with unit-determinant representatives. The boundary map is

$$\Pi:\mathcal{M}^{k,\alpha}_\Lambda(\Omega)/\mathscr D^{k+1,\alpha}(\Omega)\longrightarrow\mathcal{S}_1^{k,\alpha}(\Sigma)\times\operatorname{\mathcal{C}}^{k-1,\alpha}(\Sigma),\qquad \Pi(g)=([g^\intercal],H_g).$$

Anderson's conjecture. For each fixed $\Lambda$, the boundary map $\Pi$ is regular at generic metrics, meaning that its linearization is surjective with split kernel at those metrics.

Because Einstein metrics in dimension $3$ have constant sectional curvature, they are more rigid than higher-dimensional Einstein metrics. The conjecture asserts that this rigidity does not prevent generic regularity of the boundary data map; its resolution is not supplied in the source, so it is recorded as open.