Higher-order Bol inequality for normal solutions with bounded Q-curvature

Let $n\geq 4$ be an even integer, and let $u$ be a normal solution on $\mathbb{R}^n$ of

$$(-\Delta)^{\frac{n}{2}}u=Qe^{nu}$$

with $Q(x)\leq (n-1)!$. The normalized integrated Q-curvature is $\alpha:=\frac{2}{(n-1)!|\mathbb{S}^n|}\int_{\mathbb{R}^n}Qe^{nu}\\,\mathrm{d}x$.

Higher-order Bol inequality conjecture. One should have

$$\int_{\mathbb{R}^n}e^{nu}\\,\mathrm{d}x\geq |\mathbb{S}^n|,$$

with equality if and only if $Q(x)\equiv(n-1)!$.

This conjecture seeks a volume comparison for conformal metrics with upper-bounded Q-curvature, extending the classical two-dimensional Bol inequality to higher even dimensions. The paper establishes the assertion in radially symmetric cases and for specified polynomial Q-curvatures, while the general case remains open.