The domain conjecture for completed central reductions of enveloping algebras

Let $L$ be the coefficient field, let $\mathfrak{h}$ be a semisimple $L$-Lie algebra, let $\mathfrak{h}_0\subset\mathfrak{h}$ be an $\mathcal{O}$-Lie lattice, and let $\chi:Z(\mathfrak{h})\to L$ be an $L$-algebra homomorphism. For an integer $n$, write $\widehat{U(p^n\mathfrak{h}_0)}[1/p]_{\chi}$ for the completed enveloping algebra after inverting $p$ and taking the central reduction determined by $\chi$. Domain conjecture. For all sufficiently large $n$, the ring

$$\widehat{U(p^n\mathfrak{h}_0)}[1/p]_{\chi}$$

is a domain. This assertion is used in the paper in the study of completed enveloping algebras and their central reductions; the supplied text does not establish its status beyond the conjectural formulation.