Smooth torus quotients of Richardson varieties in the Grassmannian

Let $k$ and $n$ be positive coprime integers with $k<n$. Let $T$ denote the subgroup of diagonal matrices in $SL(n,\mathbb{C})$. We study the GIT quotient of Richardson varieties $X_w^v$ in the Grassmannian ${\mathrm{G}\mathrm{r}}_{k,n}$ by $T$ with respect to a $T$-linearised line bundle $\mathcal{L}$ corresponding to the Pl"{u}cker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety $T \backslash\mkern-6mu\backslash (X_w^v)^{ss}_T({\cal L})$ to be smooth.