Let $N(\sigma_{1}, \sigma_{2}; T)$ denote the number of nontrivial zeros of the Riemann zeta-function in the rectangle $1/2 \leq \sigma_{1} \leq \sigma_{2} < 1, 0 < Im(s) < T$.

If one could show that $N(\sigma_{1}, 1; T) \ll T$ for each $\sigma_{1} > 1/2$, then would it follow that $N(1/2, 1/2; T) \sim (1/2\pi)T \log T$?

($N(1/2, 1/2; T)$ denotes the number of zeros on the segment $\sigma = 1/2$, 0 < Im(s) < T.)