GD-degree-two conjecture for non-extreme Schubert varieties

Let $\mathcal{S}_{ij}$ be a Schubert variety in ${\rm Gr}(2,n)$. GD-degree conjecture. If $1<j-i<n-1$, then the Schubert variety $\mathcal{S}_{ij}$ has GD degree $2$. The paper reports verification for $n\leq12$, but gives no general proof in the supplied text.