Positivity determines the quantum cohomology of the odd symplectic Grassmannian of lines

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