Units of the cyclotomic completion of the integer polynomial ring

Let $\mathbb Z[q]^{\mathbb N}$ denote the cyclotomic completion of $\mathbb Z[q]$, and let $U(R)$ be the multiplicative group of units of a ring $R$. Unit-group conjecture.

$$U\bigl(\mathbb Z[q]^{\mathbb N}\bigr)=\{\pm q^i\mid i\in\mathbb Z\}.$$

The claim says that completion introduces no units beyond the signed integral powers of $q$. The supplied text does not state whether this claim is known or open.