Lieb's Wehrl norm conjecture for spin coherent states

Let ${\mathcal F}_j$ be the spin-$j$ function space on ${\mathbb S}^2={\mathbb C}\cup\{\infty\}$, and for $p\geq 1$ define the normalized norm by

$$\lVert f\rVert_p:=\left(\frac{pj+1}{\pi}\int_{\mathbb C}|f(z)|^p\frac{d^2z}{(1+|z|^2)^2}\right)^{1/p}.$$

Let $K(z,w)$ denote the coherent-vector kernel. Lieb's norm conjecture. For every $f\in{\mathcal F}_j$ and $q\geq p\geq1$,

$$\lVert f\rVert_q\leq\lVert f\rVert_p.$$

If $q>p$, equality holds if and only if $f$ is collinear with a coherent vector, namely $f(z)=cK(z,w)$ for some $c\in{\mathbb C}$ and $w\in{\mathbb C}\cup\{\infty\}$. The conjecture is the norm-inequality formulation underlying sharp Wehrl and Rényi–Wehrl entropy bounds; the supplied source records that related even-integer cases were known, while the full range, especially indices arbitrarily close to $p=2$, remained unproved there.