Wehrl-type $L^{2s}$ conjecture for affine coherent states

Fix $\alpha>0$ and $s\geq1$. Let $U$ be the affine-group representation on $L^2(\mathbb{R}_+)$, let $\eta_\alpha(k)=C(\alpha)k^\alpha e^{-k}$, and define $h_f(a,b)=\langle U(a,b)\eta_\alpha\mid f\rangle$ for normalized $f\in L^2(\mathbb{R}_+)$. Wehrl-type affine conjecture. Among all normalized $f\in L^2(\mathbb{R}_+)$,

$$\int_{-\infty}^{\infty}\int_0^{\infty}|h_f(a,b)|^{2s}a^{-2}\,\mathrm{d}a\,\mathrm{d}b$$

is maximal if and only if, up to a phase, $f=\alpha^{-1/2}U(a,b)\eta_\alpha$ for some $a>0$ and $b\in\mathbb{R}$; the maximal value is

$$\frac{2\alpha^s}{(2\alpha+1)s-1}.$$

This is the paper's general affine-group conjecture, with special cases proved there; the supplied text does not establish the full claim for every $s\geq1$.