Quadratic generation of the Parke–Taylor toric kernel

Let $\calB_n$ be the set of quadratic binomials described in Theorem toricquadratics, and let $A_n$ be the associated integer matrix. For $n e 7$,

$$I_{A_n}=\left\langle z^{u^+}-z^{u^-}:u\in\mathcal{B}_n\right\rangle:\prod_{\sigma\in\Sigma_n}z_\sigma^\infty.$$

Quadratic kernel-generation conjecture. For $n\geq 7$, the set $\calB_n$ generates $\ker_\mathbb{Z}(A_n)$. In particular, $I_{A_n}=\left\langle z^{u^+}-z^{u^-}:u\in\mathcal{B}_n\right\rangle:\prod_{\sigma\in\Sigma_n}z_\sigma^\infty$. This conjecture asserts that the quadratic binomials supplied by the preceding theorem generate the relevant toric kernel for every $n\geq 7$; the source provides no resolution or further evidence beyond its later use as a conditional hypothesis.