Quadratic generation conjecture for Richardson variety initial ideals

Let $X_w^v$ be a Richardson variety in the Grassmannian, let $G_{k,n,\ell}|_w^v$ be the corresponding restricted set of relations, and let $\operatorname{in}_{\mathbf{w}_\ell}(I(X_w^v))$ denote the initial ideal of its defining ideal with respect to $\mathbf{w}_\ell$. Assume that $G_{k,n,\ell}|_w^v$ is monomial-free. Quadratic generation conjecture. Then

$$\operatorname{in}_{\mathbf{w}_\ell}(I(X_w^v))$$

is quadratically generated. The conjecture would remove the quadratic-generation hypothesis needed in the paper's proof that the relevant initial ideal equals the kernel of the parametrization map. No resolution is given in the source.