The conjectural extension of the module-variety reconstruction theorem

Let $A$ and $B$ be graded Cohen–Macaulay $k$-algebras that are isomorphic as graded $S$-modules. In the notation of Theorem 1, let $V$ be the common finite-dimensional graded $k$-vector space such that $A$ and $B$ are isomorphic to $S\otimes_k V$ as graded $S$-modules, and consider their graded module varieties $\mathrm{Rep}_S(A,V)(k)$ and $\mathrm{Rep}_S(B,V)(k)$.

Conjecture. Theorem 1 would remain valid without any assumptions on the algebras $A$ and $B$ other than their being isomorphic as graded $S$-modules; equivalently, the conditions in that theorem would still be equivalent without assuming that $B$ is an integral domain or Gorenstein on the punctured spectrum.

The preceding theorem uses the additional hypotheses on $B$ only to prove injectivity of the natural map from $B$ to the graded endomorphism algebra of the corresponding module. The conjecture asks whether the reconstruction result nevertheless holds for arbitrary graded Cohen–Macaulay algebras with the stated graded $S$-module isomorphism.