The higher-dimensional obstruction-tensor conjecture at null infinity

Let $\{\mathscr{I},\bm{\gamma},\bm\ell,\ell^{(2)},\sigma,\mathfrak q\}$ be $\mathscr{I}$-structure data of odd dimension $\mathfrak n\ge 7$ admitting a cross-section $\iota:\Sigma \hookrightarrow\mathscr{I}$ with induced metric $h\coloneqq\iota^{\star}\bm{\gamma}$. Let $\mathcal{O}_{ab}^{\mathscr{I}}$ be the radiative obstruction tensor, and let $\mathcal{O}^{FG}$ denote the Fefferman–Graham obstruction tensor of $[h]$. Higher-dimensional obstruction-tensor conjecture. There is a constant $c_{\mathfrak n}$ depending only on $\mathfrak n$ such that

$$\iota^{\star}\big(\text{\pounds}_n^{(\frac{\mathfrak n-3}{2})} \mathcal{O}^{\mathscr{I}} \big) = c_{\mathfrak n}\,\mathcal{O}^{FG}.$$

This conjecture predicts that, in odd dimensions at least seven, sufficiently many derivatives along the null generator of the radiative obstruction tensor reproduce the Fefferman–Graham obstruction tensor of the induced conformal metric. It is motivated by the established four-dimensional relation between the derivative of the radiative obstruction and the Bach tensor, but the higher-dimensional statement remains open.