CscK metrics on rank one spherical Fano fourfolds

This four-pages note is an invitation to explore explicit K-stability for arbitrary K\"ahler classes of low dimension and low rank spherical varieties. We apply our simple combinatorial criterion of K-stability of rank one spherical varieties to the example of the blowup of the product of two copies of the projective planes along the diagonal, and obtain strong indication that it admits cscK metrics in every K\"ahler classes.