Injectivity of evaluation on infinitely many roots of unity

Let $Z$ be an infinite subset of the set of all roots of unity over $\mathbb{Q}$, and let

$$\tau ^{\mathbb Z} _{\mathbb N,Z}\colon {\mathbb Z [q]^{{\mathbb N}}}\longrightarrow P_Z(\mathbb Z)$$

be the homomorphism induced by evaluating polynomials at the elements of $Z$. Injectivity conjecture. For any infinite subset $Z$, the homomorphism $\tau ^{\mathbb Z} _{\mathbb N,Z}$ is injective. This asserts that an element of the cyclotomic completion of $\mathbb Z[q]$ is determined by its values on any infinite collection of roots of unity; the supplied text does not state whether the claim is known or open.