The quadraticity–quadratic Gröbner basis conjecture for abelian group projections

Let $G\subset \operatorname{GL}(n+1,{\bf K})$ be a finite abelian group of order $d$. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic, and $\operatorname{I}(X_{n,d}^G)$ denotes its defining ideal.

Quadraticity–quadratic Gröbner basis conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if $\operatorname{I}(X_{n,d}^G)$ has a quadratic Gröbner basis.

This conjecture asks whether quadraticity of the algebra is equivalent to the existence of a quadratic Gröbner basis for its defining ideal. It is stated among the paper's two conjectures and is motivated by the preceding results and computations; no resolution is supplied in the source.