The Effective Linear Independence conjecture for Riemann zeta zeros

Let $\{\gamma\}$ be the set of ordinates of the non-trivial zeros of the Riemann zeta function, and let $N(T)$ be the number of zeros of $\zeta(s)$ with imaginary part in $[0,T]$. For every $\varepsilon>0$, there is a positive constant $C_{\varepsilon}$, possibly ineffective, such that for every real $T\geq 2$ and every choice of integers $\ell_{\gamma}$, not all zero, satisfying $|\ell_{\gamma}|\leq N(T)$, one has

$$\left|\sum_{0<\gamma\leq T}\ell_{\gamma}\gamma\right|\geq C_{\varepsilon}e^{-T^{1+\varepsilon}}.$$

Effective Linear Independence conjecture. The above lower bound holds for all such $T$, $\varepsilon$, and integer coefficients. This is an effective quantitative strengthening of linear independence of the zero ordinates, intended to yield extreme-value results for the prime-number-theorem error term and the Möbius summatory function. The source does not specify a resolution.