Conjecture of Spira on the zeros of $\zeta^\prime(s)$

I'm looking at the review, by Haseo Ki, of Hirotaka Akatsuka, Conditional estimates for error terms related to the distribution of zeros of $\zeta'(s)$, J. Number Theory 132 (2012), no. 10, 2242–2257, MR2944752. It says,

Assuming the Riemann hypothesis, the author shows $$N(T)=N_1(T)+{T\log2\over2\pi}+O\left({\log T\over\sqrt{\log\log T}}\right)$$ and comments that there is a barrier to further improvement of this.

Shorokhodov gave an explicit counterexample to Spira's conjecture in "Pade approximates and numerical analysis of the Riemann zeta function", Computational Mathematics and Mathematical Physics, vol. 43 no. 9 (2003) pp. 1277-1298. He computed, for $T=1420$, 1000 zeros of $\zeta(s)$, 844 zeros of $\zeta^\prime(s)$, and noted $1420\log 2/2\pi\approx 156.65$.