Let $\zeta$ denote the Riemann zeta function and let denote it's complex zero. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), **but not** assuming Riemann Hypothesis ?

Assuming both the RH and the SZC, one can mimic the proof of Theorem 15.6 of Montgomery-Vaughan and show that $\frac{1}{\zeta'(\rho)} \ll X$, where $X$ is any real number $\geq |\rho| $. However, it seems this could hold assuming the SZC alone, as it appears that Montgomery-Vaughan only invoked the RH on bounding the Selberg $S(T)$ function. On the RH, it is a classical fact that $S(T) \ll \frac{\log T}{\log \log T}$, whilst $S(T) \ll \log T$ unconditionally. So the unconditional bound **seems** sufficient for the purposes of showing that $\frac{1}{\zeta'(\rho)} \ll X$ from the Montgomery-Vaughan argument.