The Linear Independence Conjecture for ordinates of zeta zeros

Consider the set of positive ordinates of non-trivial zeros of the Riemann zeta function,

$$\{\gamma\in\mathbb{R}_{>0}:\zeta(\beta+i\gamma)=0,\ \beta\in(0,1)\}.$$

Linear Independence Conjecture. This set is linearly independent over $\mathbb{Q}$. The conjecture is used in the paper's large-deviation arguments for the summatory functions under consideration; it remains open.