Asymptotic conjecture for the coefficient C_{011}(N)

Let $\operatorname{Li_2}(z)$ be the dilogarithm, let $w_0$ be the unique solution of

$$\operatorname{Li_2}(w)+2\pi i\log(w)=0,$$

with $w_0\approx 0.916198+0.182459i$, and define

$$z_0:=\frac{\log(1-w_0)}{-2\pi i}+1.$$

Thus $w_0=1-e^{-2\pi i z_0}$ and $1<\operatorname{Re}(z_0)<2$. The asymptotic conjecture for $C_{011}(N)$. One has

$$C_{011}(N)=\operatorname{Re}\left[(-2z_0e^{\pi i z_0})\frac{w_0^{-N}}{N^2}\right]+O\left(\frac{|w_0|^{-N}}{N^3}\right).$$

This gives a precise oscillatory asymptotic formula and implies that $C_{011}(N)$ does not converge. The source says that a slightly weaker version was being proved and that the main difficulty was controlling the error term.