Positivity conjecture for lower-order Q-curvatures of conformal metrics

Let $g=u^{\frac{4}{n-2m}}|dx|^2$ be a smooth conformal metric on $\mathbb{R}^n$, where $2\leq m<\frac{n}{2}$. For a real number $s$, the $2m$-order Q-curvature $Q^{(2m)}_g$ has a slow decay barrier with rate $s$ at infinity if there exists $c_0>0$ such that $Q^{(2m)}_g\geq c_0|x|^s$ for all sufficiently large $|x|$. For $1\leq k\leq m-1$, the lower-order Q-curvature $Q^{(2k)}_g$ is defined by

$$(-\Delta)^k\left(u^{\frac{n-2k}{n-2m}}\right)=Q^{(2k)}_g u^{\frac{n+2k}{n-2m}}.$$

Positivity conjecture for lower-order Q-curvatures. If $Q^{(2m)}_g$ is positive and has a slow decay barrier with rate $-2m<s\leq 0$ at infinity, then, for every $1\leq k\leq m-1$,

$$Q^{(2k)}_g>0.$$

The theorem established in the paper proves the assertion for the scalar curvature $R_g=Q^{(2)}_g$ and the fourth-order Q-curvature $Q^{(4)}_g$. The conjecture asks for positivity of all remaining lower-order Q-curvatures under the same decay assumption, beyond the cases covered by that theorem.