The Laurent polynomial Mathieu-subspace conjecture

Let $K$ be a field of characteristic zero, and let $z=(z_1,z_2,\dots,z_n)$ be $n$ commuting indeterminates. For a commutative $K$-algebra ${\mathcal A}$ and a $K$-subspace $V$ of ${\mathcal A}$, define $\widetilde V$ to be the $K$-subspace of the Laurent polynomial algebra ${\mathcal A}[z^{-1},z]$ consisting of Laurent polynomials whose constant term belongs to $V$. A Mathieu subspace (MS) is a subspace satisfying the eventual divisibility condition: if all positive powers of an element lie in it, then sufficiently high powers times any algebra element lie in it. Laurent polynomial Mathieu-subspace conjecture. The subspace $\widetilde V$ is a Mathieu subspace of ${\mathcal A}[z^{-1},z]$ if and only if $V$ is a Mathieu subspace of ${\mathcal A}$. The conjecture transfers the Mathieu-subspace property between a commutative algebra and its Laurent polynomial algebra; the supplied source presents it as the commutative case of a conjecture cited from earlier work, without providing resolution evidence.