Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

A horospherical variety is a normal $G$-variety such that a connected reductive algebraic group $G$ acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no K"{a}hler–Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be K"{a}hler–Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian $\text{SGr}(n,2n+1)$ can be arbitrarily close to zero as $n$ grows.