Egami–Matsumoto effective additive independence conjecture for zeta zero ordinates

Let $\Gamma$ be the set of imaginary parts of the non-trivial zeros of the Riemann zeta function, and let $\gamma_1,\ldots,\gamma_4\in\Gamma$. The conjecture gives a quantitative lower bound when two pairwise sums are unequal.

Egami–Matsumoto's effective independence conjecture. There exists some $\alpha<\frac{\pi}{2}$ such that either $\{\gamma_1,\gamma_2\}=\{\gamma_3,\gamma_4\}$, or

$$\left|(\gamma_1+\gamma_2)-(\gamma_3+\gamma_4)\right|\geq \exp\left(-\alpha\left(|\gamma_1|+|\gamma_2|+|\gamma_3|+|\gamma_4|\right)\right).$$

The source says that Egami and Matsumoto used this effective version to study the meromorphic continuation and natural-boundary behavior of the Goldbach generating function. It implies Fujii's additive independence conjecture, but is itself presented without a resolution, so its status remains open.