The Centralizer Conjecture for the Weyl algebra over an integral domain

Let $D$ be a commutative integral domain of characteristic zero, let $Q(D)$ be its field of fractions, and let $A_1(D)$ be the first Weyl algebra over $D$, generated by $x$ and $\partial$ with relation $[\partial,x]=1$. For elements $u,v$, write $[u,v]=uv-vu$, and let $A_1(D)^*=D^*$ denote its units.

The Centralizer Conjecture over $D$. Suppose $A,B,w\in A_1(D)$ satisfy

$$[A,B]\in A_1(D)^*=D^*$$

and

$$[A,w]=0.$$

Then

$$w\in D[A].$$

The analogous result over a field of characteristic zero implies only $w\in Q(D)[A]$ over an integral domain. This conjecture is the noncommutative Weyl-algebra analogue of the two-dimensional Centralizer Conjecture and remains open in the stated generality.