  Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a
statement about the eigenvalues of the linear operator obtained from the
quantum multiplication of the anticanonical class of $X$. Conjecture
$\mathcal{O}$ is a conjecture that Property $\mathcal{O}$ holds for any Fano
variety. Pasquier listed the smooth non-homogeneous horospherical varieties of
Picard rank 1 into five classes. Conjecture $\mathcal{O}$ has already been
shown to hold for the odd symplectic Grassmannians which is one of these
classes. We will show that Conjecture $\mathcal{O}$ holds for two more classes
and an example in a third class of Pasquier's list. The theory of
Perron-Frobenius reduces our proofs to be graph-theoretic in nature.
