Conjectural decomposition of critical extrinsic Q-curvature

Let $M^n$ be a hypersurface embedded in a Riemannian manifold $X$, with induced metric $h$ and ambient metric $g$. Write ${\bf Q}_n(g)$ for the critical extrinsic $Q$-curvature, $Q_n(h)$ for the intrinsic critical $Q$-curvature of $(M,h)$, and call a scalar a local conformal invariant of the embedding when it has the appropriate conformal covariance under changes of the ambient metric. A total divergence is the divergence of a vector field on $M$. For even $n$, let the Pfaffian of $(M,h)$ denote its intrinsic Pfaffian curvature.

Conjectural decomposition. If $n$ is even, then ${\bf Q}_n(g)$ is a linear combination of $Q_n(h)$, local conformal invariants of the embedding $M\hookrightarrow X$, and a total divergence. Equivalently, using the Deser–Schwimmer decomposition of $Q_n(h)$, it is a linear combination of the Pfaffian of $(M,h)$, local conformal invariants of the embedding, and a total divergence. If $n$ is odd, then ${\bf Q}_n(g)$ is a linear combination of local conformal invariants of the embedding $M\hookrightarrow X$ and a total divergence.

This conjecture proposes a higher-dimensional analogue of the critical four-dimensional decomposition, separating the extrinsic $Q$-curvature into intrinsic topological, conformally invariant, and divergence contributions. The source gives no resolution, so the conjectural decomposition remains open.