Montgomery's simplicity conjecture for the zeros of the Riemann zeta-function

Let $N^*(T)$ be the multiplicity-weighted zero count

$$N^*(T):=\sum_{\substack{\rho\\0<\gamma\leq T}}m_\rho=\sum_{\substack{\rho\ \text{distinct}\\0<\gamma\leq T}}m_\rho^2,$$

where $m_\rho$ is the multiplicity of the zero $\rho$. Montgomery's simplicity conjecture. As $T\to\infty$,

$$N^*(T)=TL+o(TL).$$

Since $N(T)\sim TL$, the assertion predicts that asymptotically all non-trivial zeros are simple. It is presented as a consequence of a stronger pair-correlation conjecture together with the Riemann Hypothesis, but remains unproved.