Reduced resonance scheme conjecture for two-step nilpotent Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over $\mathbb{k}$, and set $A=\operatorname{CE}(\mathfrak{g})$. Let $S$ be the symmetric algebra governing the Koszul module $\mathfrak{B}_1(H^*(A))$. Reduced resonance scheme conjecture. (i) The resonance scheme $\operatorname{Spec} S/\operatorname{Ann}_S\mathfrak{B}_1(H^*(A))$ is reduced. (ii) Its support $\mathcal{R}^1(H^*(A))$ is a union of linear subspaces of $H^1(A)$. The surrounding discussion says that computational evidence suggests this claim for $2$-step nilpotent $\mathfrak{g}$; the displayed statement itself is broader, and the supplied text gives no resolution status.