Fefferman–Graham flatness conjecture for Poincaré–Einstein representatives

Let $(M^d,\boldsymbol{c})$ with $d$ even be a Fefferman–Graham-flat conformal manifold whose boundary satisfies

$$\mathring{\rm{ I\hspace{-.2mm} I}} = {\mathring{{\bf\rm I\hspace{-.2mm} I \hspace{-.2mm} I}}{\hspace{.2mm}}}{} = \cdots = \mathring{\underline{\overline{\rm{d/2}}}} = 0.$$

Fefferman–Graham flatness conjecture. There exists a metric representative in $(M_+,\boldsymbol{c})$ that is formally to all orders Poincaré–Einstein.

This conjecture proposes a higher-dimensional extension of the Fefferman–Graham flat problem: the vanishing of the successive trace-free second fundamental forms should allow the construction of a formally Poincaré–Einstein representative. The source notes that the case $d=6$ has been examined, while the general statement remains unresolved.