  Let $\mbox{IG}:=\mbox{IG}(2,2n+1)$ denote the odd symplectic Grassmannian of
lines which is a horospherical variety of Picard rank 1. The quantum cohomology
ring $\mbox{QH}^*(\mbox{IG})$ has negative structure constants. For $n \geq 3$,
we give a positivity condition that implies the quantum cohomology ring
$\mbox{QH}^*(\mbox{IG})$ is the only quantum deformation of the cohomology ring
$\mbox{H}^*(\mbox{IG})$ up to the scaling of the quantum parameter. This is a
modification of a conjecture by Fulton.
