The bounded-distance characterization of mock characters

Let $\a0\mathbb U$ denote the complex unit disc, and let $\a0\mathbb D(f,g;y)$ be the pseudometric on completely multiplicative functions defined by

$$\mathbb D(f,g;y):= \left(\sum_{p \leq y} \frac{1-\operatorname{Re} f(p) \overline{g(p)}}{p}\right)^{1/2}.$$

A mock character is a completely multiplicative function whose sequence is automatic but not eventually periodic and which vanishes precisely at zero and at integers sharing a common factor with a fixed integer. A Dirichlet character is understood in the usual sense.

Bounded-distance conjecture. For any mock character $\kappa$, there exists a Dirichlet character $\chi$ such that $\mathbb D(\kappa,\chi;y)$ is bounded. Conversely, if $\kappa: \mathbb Z \to \mathbb U$ is completely multiplicative and at a bounded distance from some Dirichlet character, then $\kappa$ must be a mock character.

The proposition immediately preceding this conjecture proves the first assertion for the family $\kappa_a$ considered in the paper. The conjecture proposes the corresponding statement for all mock characters and a converse characterization; the source gives no resolution status.