8D conformal gravity with Einstein sector, and its relation to the Q-curvature

We first streamline the construction of the unique six-dimensional conformal gravity action found by L\"u, Pang and Pope, that admits Einstein metrics as solutions to the field equations. We then prove that there exists a unique eight-dimensional conformal gravity action that admits Einstein metrics as solutions to the field equations, and explicitly build the corresponding action. Finally, we relate these results to Branson's Q-curvature and the Fefferman-Graham obstruction tensor, to conclude that on every even-dimensional space there exists a unique -- up to boundary terms -- conformally-invariant gravity theory that is extremised by Einstein metrics.