Universal formula for the eighth-order GJMS-operator and Q-curvature

Let $(M,g)$ be a locally conformally flat manifold of dimension $n\geq3$. Let $P_2,P_4,P_6,P_8$ be the GJMS-operators, let $Q_8$ be the eighth-order Q-curvature, and let ${\sf P}$ be the Schouten tensor. Define

$${\mathcal P}_8=(3P_2P_6+3P_6P_2+9P_4^2)-(8P_2P_4P_2+12P_2^2P_4+12P_4P_2^2)+18P_2^4.$$

Eighth-order recursive formula.

$$P_8={\mathcal P}_8^0-3!4!2^3\delta({\sf P}^3\#d)+\left({n\over2}-4\right)Q_8,$$

with

$$Q_8={\mathcal Q}_8-12(Q_6-{\mathcal Q}_6)Q_2-18(Q_4-{\mathcal Q}_4)^2+4!3!2^7v_8,$$

where $v_8=2^{-4}\operatorname{tr}(\wedge^4{\sf P})$. This extends the established locally conformally flat eighth-order construction to non-critical dimensions and asserts that the resulting conformally covariant operator equals $P_8$; the general statement remains open.