Quadratic generation of the Parke–Taylor ideal

Let $\calI(\mathrm{PT}_n^\circ)$ denote the Parke–Taylor ideal before saturation, let $\calF$ be the lifts of the Plücker relations, and let $\calB_n$ be the quadratic binomials from Theorem toricquadratics. Work in the Laurent polynomial ring $\mathbb{C}[z_\sigma^{\pm}]$.

Quadratic Parke–Taylor generation conjecture. The ideal $\tilde{\mathcal{I}}(\mathrm{PT}_n^\circ)\subseteq\mathbb{C}[z_\sigma^{\pm}]$ is generated by the quadratics in $\calF$ and $\calB_n$.

The conjecture would give a quadratic generating set for the Laurent version of the Parke–Taylor ideal, combining lifted Plücker relations with the toric quadratic binomials. The supplied text gives no resolution or status evidence.