In 2012 E. H. Lieb and J. P. Solovej proved a substantial part of this conjecture, confirming the minimum value of the Wehrl entropy for Bloch coherent states, and the fact that it is reached for any pure Bloch coherent state. The uniqueness of the minimizers was proved in 2022 by R. L. Frank and A. Kulikov, F. Nicola, J. Ortega-Cerda' and P. Tilli.