The strong spacing conjecture for transformed zeta zeros

Let $\lambda_j=\eta_j^2$ be the transformed nontrivial zeta zeros, ordered as in the source. For a parameter $\gamma$ with

$$\gamma\in[0,1/3),$$

consider consecutive transformed zeros that are not conjugates and with $\operatorname{Im}(\lambda_j)\ne0$. Strong spacing conjecture. There exists a constant $C>0$, depending on $\gamma$, such that

$$\operatorname{Re}(\lambda_{j+1}-\lambda_j)\geq C\,\operatorname{Re}(\lambda_j)^\gamma$$

for every such $j$. This generalizes the predicted strong spacing regime suggested by quadratic repulsion from random matrix theory; the source does not report a proof or disproof.