Asymptotic conjecture for the coefficient C_{121}(N)

Let $w_0\approx 0.916198+0.182459i$ be the unique solution of $\operatorname{Li_2}(w)+2\pi i\log(w)=0$, and define

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

Then $w_0=1-e^{-2\pi i z_0}$ and $1<\operatorname{Re}(z_0)<2$. The asymptotic conjecture for $C_{121}(N)$. One has

$$C_{121}(N)=\operatorname{Re}\left[-z_0\sqrt{2e^{\pi i z_0}\left(e^{\pi i z_0}+(-1)^N\right)}\frac{w_0^{-N/2}}{N^2}\right]+O\left(\frac{|w_0|^{-N/2}}{N^3}\right).$$

Together with the corresponding conjecture for $C_{011}(N)$, this predicts exponentially growing oscillations and failure of convergence for these coefficients. The source presents this as a conjectural asymptotic formula.