The realizability criterion for complete intersection rings

Assume that a local ring $R$ has minimal regular presentation $Q/I$, let $k$ be its residue field, and fix minimal generators $f_1,\ldots,f_n$ of $I$. Let

$$E=Q[e_1,\ldots,e_n\mid \partial e_i=f_i]$$

and let

$$\mathcal{S}=k[\chi_1,\ldots,\chi_n],$$

where each $\chi_i$ has cohomological degree $2$. For an $R$-complex $M$ with finitely generated homology, write ${\rm V}_R(M)$ for its support variety in ${\operatorname{Spec}^*}\mathcal{S}$. Realizability conjecture. The ring $R$ is complete intersection if and only if $\\{\bm{0}\\}$ is realizable as ${\rm V}_R(M)$ for some $R$-complex $M$ with $\operatorname{H}(M)$ finitely generated. This proposes a converse to the known obstruction: over Cohen–Macaulay rings that are not complete intersections, every such complex with nonzero finitely generated homology has support variety of dimension at least $2$, so the origin is not realizable. The complete-intersection direction follows from the established realizability of every closed subset, while the converse remains open in the stated generality.