The cylindrical-dilatation Maxwellian asymptotics conjecture

Let $B$ be a collision cross-section satisfying condition with homogeneity $\gamma<\gamma_{\mathrm{crit}}=-\frac32$, and let $g(t,w)$ be a weak solution of the cylindrical-dilatation Boltzmann equation with shear parameter $K$. Let $\beta(t)$ be the temperature scale. Cylindrical-dilatation Maxwellian asymptotics conjecture. There exists a weak solution such that

$$g\left(t,\frac{\xi}{\sqrt{\beta(t)}}\right)\longrightarrow C_0e^{-\lvert\xi\rvert^2}$$

in $L^2(\mathbb{R}^3;e^{-\lvert\xi\rvert^2}d\xi)$ as $t\to\infty$, for some $C_0>0$, and

$$\beta(t)=Ct^{\frac43}$$

as $t\to\infty$. The claim is presented as a consequence expected from the general Hilbert-expansion strategy for the regime $\gamma<\gamma_{\mathrm{crit}}$; it is not proved in the source.