Equivariant compactifications of a unipotent group by a smooth projective horospherical variety of Picard number one

Cheong proved that if $S$ is a simple Lie group and $P$ is a parabolic subgroup, then $S/P$ admits a unique equivariant compactification of the unipotent radical $N$ of $P$, up to isomorphism, provided that $S/P$ is not isomorphic to a projective space. We generalize this result to a smooth projecti