  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.
