Helson's zero-free conjecture for twisted Beurling Dirichlet series

Let $\mathbb{N}_q$ be a Beurling system satisfying Bohr's condition, let $f\in\mathcal{H}_q^2$ be outer, meaning that

$$\{fg:\,g\in\mathcal{H}_q^\infty\}$$

is dense in $\mathcal{H}_q^2$, and let $f_\chi$ denote the twisted Dirichlet series associated with $f$ and a completely multiplicative character $\chi$.

Helson's conjecture. The function $f_\chi$ never has any zeros in its half-plane of convergence.

This is a special case of a conjecture made by Helson concerning twisted Dirichlet series. The claim relates outerness in the Hardy space of a Beurling system to zero-freeness of almost-everywhere twists, but the source gives no resolution status.