Rubinstein–Sarnak linear independence hypothesis for Dirichlet $L$-functions

Fix a modulus $q$, and consider the zeros of Dirichlet $L$-functions attached to primitive characters modulo $q$. For a zero $\varrho=\beta+i\gamma$, call $\gamma$ its imaginary part. Linear Independence hypothesis (LI). The imaginary parts of the zeros of all Dirichlet $L$-functions attached to primitive characters modulo $q$ are linearly independent over $\mathbb{Q}$. This hypothesis is used by Rubinstein and Sarnak in their analysis of prime-number races and the existence of limiting race distributions. It remains unproved in the source.