Lieb's Wehrl entropy lower-bound conjecture for spin states

Let ${\mathcal F}_j$ be the spin-$j$ function space, let $\lVert\cdot\rVert_2$ be the normalized norm, and let

$$S_j(|f|^2):=-\frac{2j+1}{\pi}\int_{\mathbb C}|f(z)|^2\ln|f(z)|^2\frac{d^2z}{(1+|z|^2)^2}$$

be the Wehrl entropy of an $f\in{\mathcal F}_j$ normalized by $\lVert f\rVert_2=1$. Lieb's Wehrl entropy conjecture. For every such $f$,

$$S_j(|f|^2)\geq\frac{2j}{2j+1},$$

and equality holds whenever $f$ is, up to a unimodular constant, a coherent vector. This bound would follow by endpoint differentiation from the preceding norm or Rényi–Wehrl conjecture. The source says that the relevant even-integer cases were known, but that the argument near $q=2$ was not available there; consequently the full entropy bound remained conjectural in the source.