Shifting the ordinates of zeros of the Riemann zeta function

Let y≠ 0 and $C>0$. Under the Riemann Hypothesis, there is a number T_*>0 $($depending on y and $C)$ such that for every T≥ T_*, both \[ ζ(12+iγ)=0 ζ(12+i(γ+y))≠ 0 \] hold for at least one γ in the interval [T,T(1+ε)], where ε:=T-C/loglog T.