Shifting the ordinates of zeros of the Riemann zeta function

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for at least one $\gamma$ in the interval $[T,T(1+\epsilon)]$, where $\epsilon:=T^{-C/\log\log T}$.
Comment: 14 pages; comments are welcome