Generalized Fefferman-space reconstruction conjecture

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$. Suppose the Weyl tensor satisfies

$$\widetilde{\mathsf{W}}_{a b c d} \widetilde{k}^{a} \widetilde{v}^{b} \widetilde{k}^{c}=0,\qquad \widetilde{\mathsf{W}}_{a b c d} \widetilde{k}^{a} \widetilde{v}^{b} \widetilde{\nabla}^{c} \widetilde{k}^{d}=0$$

for any $\widetilde{k}\in\Gamma(\widetilde{K})$ and $\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$, and suppose the Fefferman--Graham obstruction tensor satisfies

$$\widetilde{\mathsf{F}\mkern-4mu\mathsf{G}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}=0$$

for any $\widetilde{k}\in\Gamma(\widetilde{K})$. Generalized Fefferman-space reconstruction conjecture. Then $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ is locally conformally isometric to a perturbed Fefferman space $(\widetilde{\mathcal{M}}',\widetilde{\mathbf{c}}'_{\widetilde{\xi}},\widetilde{k}')\longrightarrow(\mathcal{M},H,J)$, and the perturbation one-form $\widetilde{\xi}$ is determined by the CR data $\left(\bm{\xi}_{\alpha}^{(0)},[\nabla,\bm{\xi}_{0}^{(0)}],\bm{\xi}_{0}^{(2k)}\right)_{k=1,\ldots,m+1}$, where

$$\bm{\xi}_{0}^{(0)}=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\bm{\xi}^{\alpha}_{(0)}-\nabla^{\alpha}\bm{\xi}_{\alpha}^{(0)}\right).$$

This is proposed as a higher-dimensional generalization of a four-dimensional integration result for perturbed Fefferman spaces. The source does not provide evidence that the conjecture has been proved or disproved.