The two-dimensional Centralizer Conjecture over an integral domain

Let $D$ be a commutative integral domain of characteristic zero, and let $A,B,w\in D[x,y]$. Define

$$\operatorname{Jac}(u,v)=u_xv_y-u_yv_x.$$

If $\operatorname{Jac}(A,B)\in D^*$, then $B$ is a Jacobian mate of $A$; the Jacobian centralizer of $A$ consists of the elements $w$ satisfying $\operatorname{Jac}(A,w)=0$.

The two-dimensional Centralizer Conjecture over $D$. Suppose

$$\operatorname{Jac}(A,B)\in D[x,y]^*=D^*$$

and

$$\operatorname{Jac}(A,w)=0.$$

Then $w\in D[A]$.

The preceding theorem proves only $w\in Q(D)[A]$, where $Q(D)$ is the field of fractions of $D$. The conjecture asserts that no denominators are needed, a conclusion known over fields of characteristic zero but open for general integral domains.