In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes\documentclass[12pt]{minimal}
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				\begin{document}$$( \mathscr {M}, g )$$\end{document}(M,g)with conformal boundary\documentclass[12pt]{minimal}
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				\begin{document}$$( \mathscr {I}, \mathfrak {g})$$\end{document}(I,g). We establish a correspondence, near\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {I}$$\end{document}I, between such spacetimes and their conformal boundary data on\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {I}$$\end{document}I. More specifically, given a domain\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D} \subset \mathscr {I}$$\end{document}D⊂I, we prove that the coefficients\documentclass[12pt]{minimal}
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				\begin{document}$$\mathfrak {g}^{\scriptscriptstyle (0)} = \mathfrak {g}$$\end{document}g(0)=gand\documentclass[12pt]{minimal}
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				\begin{document}$$\mathfrak {g}^{\scriptscriptstyle (n)}$$\end{document}g(n)(theundetermined term, orstress energy tensor) in a Fefferman–Graham expansion of the metricgfrom the boundary uniquely determinegnear\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}D, provided\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}Dsatisfies ageneralised null convexity condition(GNCC). The GNCC is a conformally invariant criterion on\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}D, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {M}$$\end{document}Mnear\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}D, and with the pseudoconvexity degenerating in the limit at\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}D. As a corollary of this result, we deduce that conformal symmetries of\documentclass[12pt]{minimal}
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				\begin{document}$$( \mathfrak {g}^{\scriptscriptstyle (0)}, \mathfrak {g}^{\scriptscriptstyle (n)} )$$\end{document}(g(0),g(n))on domains\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D} \subset \mathscr {I}$$\end{document}D⊂Isatisfying the GNCC extend to spacetime symmetries near\documentclass[12pt]{minimal}
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				\begin{document}$$\mathscr {D}$$\end{document}D. The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary.