Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII

The wonderful compactification $X_m$ of a symmetric homogeneous space of type AIII$(2,m)$ for each $m\ge 4$ is Fano, and its blowup $Y_m$ along the unique closed orbit is Fano if $m\ge 5$ 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 Kähler-Einstein metric for each $m\ge 4$ and $Y_m$ admits a Kähler-Einstein metric if and only if $m=4,5$.