Conjecture on kinetic equations for scale-invariant potentials

Let $\Phi$ be a scale-invariant potential of the form

$$\Phi(x)=\frac{1}{|x|^s},\qquad s>\frac12,$$

and consider the scaling

$$\Phi_{\varepsilon}(x)=\varepsilon\Phi(x),\qquad N=\varepsilon^{-\gamma},$$

with $\gamma>0$. Scale-invariant potentials conjecture. If $s\in(\frac12,1)$, then the one-particle marginal $f^\varepsilon_1$ converges to a solution of a kinetic equation when

$$\gamma=\frac{4}{1+s},$$

and the limiting equation is the Balescu–Lenard equation. If $s=1$, then on the logarithmic time scale $\bar t=|\log\varepsilon|t$, $f^\varepsilon_1$ converges to a solution of the Landau equation. If $s>1$, then a kinetic limit is obtained when

$$\gamma=\frac{2}{s},$$

and the limiting equation is the Boltzmann equation. These cases propose the kinetic limits for the three regimes of the scale-invariant interaction, but the source does not specify which assertions have been proved.