Stable rationality conjecture for quotients by free central extensions

Let $A$ be a finite abelian group, let $F^c(A)$ be its free central extension, and let $V$ be a faithful linear representation of $F^c(A)$ over an algebraically closed field $k$. Free-central-extension quotient conjecture. The quotient $V/F^c(A)$ is stably rational. The claim concerns a stable-rationality strengthening of rationality results for quotients by central extensions; the supplied text gives no resolution.