Centre-determines-cancellation conjecture for affine noetherian domains

Let $k$ be an uncountable algebraically closed field of characteristic zero, and let $A$ be an affine noetherian domain over $k$. Write $Z(A)$ for the centre of $A$. An algebra is cancellative if its isomorphism type is determined after adjoining a commutative polynomial variable, and strongly cancellative if the analogous determination holds after adjoining any finite number of such variables. Suppose that $Z(A)$ is affine and cancellative, respectively strongly cancellative. Centre-determines-cancellation conjecture. Then $A$ is cancellative, respectively strongly cancellative. This conjecture proposes that, over sufficiently well-behaved fields, cancellation properties of an affine noetherian domain are completely controlled by its centre; the source gives no resolution.