Existence and uniqueness conjecture for G-twisted Jacobian algebras

Let $f$ be an invertible polynomial and let $G$ be a commutative group acting as in the preceding definitions. Write ${\rm Jac}'(f,G)$ for a $G$-twisted Jacobian algebra of $f$, and let $\left(\mathrm{Jac}'(f,G)\right)^G$ denote its $G$-invariant subalgebra. Existence and uniqueness conjecture. (a) A $G$-twisted Jacobian algebra ${\rm Jac}'(f,G)$ of $f$ should exist. (b) The subalgebra $\left(\mathrm{Jac}'(f,G)\right)^G$ should be uniquely determined by $(f,G)$ up to isomorphism. The conjecture addresses both existence of the algebraic structure satisfying the stated axioms and uniqueness of its invariant subalgebra; the supplied text gives no evidence of resolution.