Dirichlet ordinates in arithmetic progressions

Given a Dirichlet character $\chi$, call a real number $\gamma$ a $\chi$-ordinate if $L(\sigma+i\gamma,\chi)=0$ for some $0<\sigma<1$. Let $a$ and $b$ be positive real numbers. The arithmetic-progression conjecture.

1. For any Dirichlet character $\chi$, at most two points in $\{a+kb:k\in\mathbb Z\}$ are $\chi$-ordinates; when $a=0$, at most one point in $\{kb:k\in\mathbb Z\}$ is a $\chi$-ordinate.
2. No real number is a $\chi$-ordinate for two different Dirichlet characters $\chi$.

The first assertion follows formally from the linear-independence conjecture because three distinct arithmetic-progression elements are rationally dependent, while the second uses the multiset interpretation of Dirichlet ordinates. The source treats these as consequences of the main linear-independence conjecture rather than independent evidence, and the relevant general assertions remain open.