The irreducible-smooth characteristic scheme conjecture

Let $\mathcal{C}=(\mathbb{Z}/2)\langle a_1,\ldots,a_n\rangle/I$ be the characteristic algebra, and let $X$ be the affine scheme in $\mathbb{A}^n$ over $\mathbb{Z}/2$ obtained from its abelianization. A $(\mathbb{Z}/2)$-rational point of $X$ is a point whose coordinates corresponding to nonzero-degree generators vanish. Irreducible-smooth characteristic scheme conjecture. The scheme $X$ is irreducible and smooth at each $(\mathbb{Z}/2)$-rational point. The source says this has been verified in many examples and that it would imply the augmentation-independence conjecture for the first-order Poincaré–Chekanov polynomial; no general proof is given.