The three-dimensional restriction conjecture for quadratic monomial projection algebras

Let $n\ge 3$ and take a cyclic group $G\subset \operatorname{GL}(n+1,{\bf K})$ of order $d$ generated by a diagonal matrix

$$\left(\begin{array}{lllllll} e^{\alpha_0} & 0 & \cdots & 0\\\\ 0 & e^{\alpha_1} & \cdots & 0\\\\ \vdots & \vdots & \ddots & \vdots\\\\ 0 & 0 & \cdots & e^{\alpha_n}\end{array}\right)$$

where $e$ is a $d$-th primitive root of $1\in {\bf K}$. For $i<j<k$, let $G^{i,j,k}\subset \operatorname{GL}(3,{\bf K})$ be the cyclic group generated by

$$\left(\begin{array}{lllllll} e^{\alpha_i} & 0 & 0\\\\ 0 & e^{\alpha_j} & 0\\\\ 0 & 0 & e^{\alpha_k}\end{array}\right).$$

The three-dimensional restriction conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if, for all $i<j<k$, the algebra $\operatorname{A}(X_{2,d}^{G^{i,j,k}})$ is quadratic.

This conjecture proposes that quadraticity of these algebras for cyclic diagonal group actions in arbitrary dimension is detected by all three-dimensional coordinate restrictions. It is motivated by the preceding results, a cited corollary, and Macaulay2 computations; no resolution is supplied in the source.