Cheng–Huang–Li integral decay conjecture for the Lane–Emden system

Let $(u,v)$ solve the Lane–Emden system in $\mathbb{R}^n$ with $p\geq q$, and let $\beta$ denote the exponent used in the source's integral estimate. Cheng–Huang–Li conjecture. There exists $s>0$ such that $n-s\beta<1$ and

$$\int_{B_R}v^s\leq C R^{n-s\beta}.$$

The source states that this conjecture is equivalent to the Lane–Emden conjecture. Since the full Lane–Emden nonexistence problem remains open in higher dimensions, this integral estimate remains open in general.