Nonpositive-curvature local diffeomorphism conjecture for cusp metrics

Let $(N,g_{0})$ be a Poincaré–Einstein metric with cusps, let $K_{N}$ denote its sectional curvature, and let $\Phi^{g_{0}}$ be the Einstein-equation map defined near $g_{0}$. Let $\Pi$ be the boundary map from Poincaré–Einstein metrics with cusps on $N$ to the space ${\mathcal C}$ of conformal classes at infinity.

Nonpositive-curvature local diffeomorphism conjecture. If

$$K_{N}\leq 0,$$

then $\Phi^{g_{0}}$ is a submersion at $g_{0}$, and $\Pi$ is a diffeomorphism in a neighborhood of $g_{0}$.

This conjecture concerns the local deformation theory of cusp Poincaré–Einstein metrics and would imply that their conformal infinities locally determine the metrics smoothly. The supplied text gives no resolution.