Consistency criterion for torsion-free nilpotent group presentations

Let $n\in\mathbb N$, let $T$ be a sequence of $\binom{n}{3}$ indeterminates, and let $C_1(T),\ldots,C_r(T)$ be the coefficients obtained from the associativity defect polynomial defined by the multiplication polynomials $F(T;x,y)$. Define

$$\mathcal C_n=\{t\in\mathbb Z^{\binom{n}{3}}\mid G(t)\text{ is consistent}\}.$$

Consistency conjecture. If

$$t\in\{t\in\mathbb Z^{\binom{n}{3}}\mid C_1(t)=\cdots=C_r(t)=0\},$$

then $t\in\mathcal C_n$.

Associativity proves the necessity of the coefficient equations; the conjecture asserts their sufficiency and would give an alternative algebraic description of consistency for these nilpotent presentations. The supplied text reports only experimental evidence and does not indicate a resolution.