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

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