Linear independence of Riemann zeta zero ordinates

Let $\zeta(s)$ be the Riemann zeta function, and consider the positive imaginary parts of its nontrivial zeros. The zeta linear-independence conjecture. These imaginary parts are linearly independent over the rationals. This is presented as the older conjecture generalized by the Dirichlet $L$-function version; the source reports numerical evidence for the smallest twenty ordinates but says that direct theoretical evidence remains lacking.