  The wonderful compactification $X_m$ of a symmetric homogeneous space of type
AIII$(2,m)$ for each $m \geq 4$ is Fano, and its blowup $Y_m$ along the unique
closed orbit is Fano if $m \geq 5$ and Calabi-Yau if $m = 4$. Using a
combinatorial criterion for K-polystability of smooth Fano spherical varieties
obtained by Delcroix, we prove that $X_m$ admits a K\"ahler-Einstein metric for
each $m \geq 4$ and $Y_m$ admits a K\"ahler-Einstein metric if and only if $m =
4, 5$.
