Pointwise positivity conjecture for the second- and fourth-order curvature quantities

Let $n>6$ and let $u\in \mathcal{C}^{\infty}(\mathbb{R}^n\setminus\\{0\\})$ be a positive solution of the sixth-order limiting equation referenced in the source. Define

$$Q_2(u):=-\Delta u-\frac{4}{n-6}\frac{|\nabla u|^2}{u}.$$

For the conformally flat metric $g=u^{4/(n-6)}\delta$, define $Q_4(u)$ by

$$\begin{aligned} Q_4(u):={}&\Delta^2 u-\frac{8}{n-6}u^{(8-n)/2}\langle\nabla u,\nabla\Delta u\rangle-\frac{4}{n-6}u^{(8-n)/2}|D^2u|^2\\\\ &-\frac{8(8-n)}{(n-6)^2}u^{7-n}D^2u(\nabla u,\nabla u)-\frac{4(8-n)}{(n-6)^2}u^{7-n}|\nabla u|^2\Delta u\\\\ &-\frac{2(n-7)(n-8)}{(n-6)^3(n-4)}u^{(20-3n)/2}|\nabla u|^4. \end{aligned}$$

Pointwise positivity conjecture. The estimates

$$Q_2(u)\geq\sqrt{\frac{n-6}{n}}u^{n/(n-6)},\qquad Q_4(u)\geq\sqrt{\frac{n-6}{n}}u^{n/(n-6)}$$

hold throughout $\mathbb{R}^n\setminus\\{0\\}$. In particular, the curvatures $Q_g^2$ and $Q_g^4$ associated with $g$ are both positive.

The claim is posed after observing that the assumptions $Q_g^2\geq0$ and $Q_g^4\geq0$ do not directly imply $\Delta^2u\geq0$. Its status is not resolved in the supplied material.