  Vanishing of the Fefferman-Graham obstruction tensor was used by Andersson
and Chru{\'s}ciel to show stability of the asymptotically de Sitter spaces in
even dimensions. However, existing proofs of hyperbolicity of this equation
contain gaps. We show in this paper that it is indeed a well-posed hyperbolic
system with unique up to diffeomorphism and conformal transformations smooth
development for smooth Cauchy data. Our method applies also to equations
defined by various versions Graham-Jenne-Mason-Sparling operators. In
particular, we use one of these operators to propagate Gover's condition of
being almost conformally Einstein. This allows to study initial data also for
Cauchy surfaces which cross conformal boundary. As a by-product we show that on
globally hyperbolic manifolds one can always choose conformal factor such that
Branson Q-curvature vanishes.
