The boundary-image conjecture for Poincaré–Einstein metrics

Let ${\mathcal E}$ be the space of smooth Poincaré–Einstein metrics on $M$, let $\bar{\mathcal E}$ be its closure, and let

$$\bar{\Pi}:\bar{\mathcal E}\rightarrow {\mathcal C}$$

be the continuous extension of the boundary map, where ${\mathcal C}$ is the space of conformal classes at infinity. For a component ${\mathcal E}_{0}$ of ${\mathcal E}$, write $\partial{\mathcal E}_{0}$ for its boundary in $\bar{\mathcal E}$.

Boundary-image conjecture. For any component ${\mathcal E}_{0}$ of ${\mathcal E}$, $\bar{\Pi}(\partial {\mathcal E}_{0})$ has empty interior in ${\mathcal C}$.

The conjecture is motivated by the fact that $\Pi$ has Fredholm index zero. It would constrain how orbifold-singular and cusped limits can occur within components of the moduli space, but its status is not resolved in the supplied text.