The wonderful compactification Xmβ of a symmetric homogeneous space of type
AIII(2,m) for each mβ₯4 is Fano, and its blowup Ymβ along the unique
closed orbit is Fano if mβ₯5 and Calabi-Yau if m=4. Using a
combinatorial criterion for K-stability of smooth Fano spherical varieties
obtained by Delcroix, we prove that Xmβ admits a K\"ahler-Einstein metric for
each mβ₯4 and Ymβ admits a K\"ahler-Einstein metric if and only if m=4,5.Comment: 20 pages, 6 figures. arXiv admin note: text overlap with
arXiv:2011.12551, arXiv:2203.1105