Suppose $a,b\in (0,1)$. I'm interested in comparison of an asymptotic behavior of ${\mathrm{Li}}_{-n}(a)$ and ${\mathrm{Li}}_{-n}(b)$ for $n\to \mathrm{\infty}$.

Such functions exhibit approximately factorial-like (faster than exponential) growth rate. The particular case ${\mathrm{Li}}_{-n}\phantom{\rule{negativethinmathspace}{0ex}}\left({\textstyle \frac{1}{2}}\right)$ for $n\ge 1$ gives (up to a coefficient) a combinatorial sequence called Fubini numbers or orderded Bell numbers[1][2][3] (number of outcomes of a horse race provided that ties are possible). This sequence is known to have the following asymptotic behavior:

$$\begin{array}{}\text{(1)}& {\mathrm{Li}}_{-n}\phantom{\rule{negativethinmathspace}{0ex}}\left({\textstyle \frac{1}{2}}\right)\sim \frac{n!}{{\mathrm{ln}}^{n+1}2}.\end{array}$$

After some numerical exprerimentation I conjectured the following behavior:

$$\begin{array}{}\text{(2)}& \mathrm{ln}\phantom{\rule{negativethinmathspace}{0ex}}\left(\frac{{\mathrm{Li}}_{-n}(a)}{{\mathrm{Li}}_{-n}(b)}\right)=(n+1)\cdot \mathrm{ln}\phantom{\rule{negativethinmathspace}{0ex}}\left(\frac{\mathrm{ln}b}{\mathrm{ln}a}\right)+o\phantom{\rule{negativethinmathspace}{0ex}}\left({n}^{-N}\right)\end{array}$$

for arbitrarily large N (so, the remainder term decays faster than any negative power of n). It looks like the remainder term is oscillating with exponentially decreasing amplitude, but I haven't yet found the exact exponent base or asymptotic oscillation frequency.

Could you suggest a proof of (2) or further refinements of this formula?