Gonek–Hejhal conjecture at k=1

Let $\rho=\beta+i\gamma$ range over non-trivial zeros of the Riemann zeta function $\zeta(s)$, and define

$$J_1(T):=\sum_{0<\gamma\leq T}\frac{1}{|\zeta'(\rho)|^2}.$$

Gonek–Hejhal conjecture at $k=1$. One has

$$J_1(T)\ll T.$$

This is the special case needed in the paper's arguments and is presented there as a simpler conjectural assumption than the full Gonek–Hejhal conjecture. Its status is open.