Is $$\lim_{x\to\infty} \frac{e^{(\ln ax)^k}}{e^{(\ln x)^k}}$$ finite for all positive real $a$ and $k$?
I have tried this on Desmos with $a$ and $k$ less than 2 where it seems to converge after around $10^{13}$, but with larger values the function is essentially a vertical line, though I'm not sure if it eventually converges.
This would answer if $e^{(\ln x)^k}$ (a quasi-polynomial time function) is regularly varying. Here is the Wikipedia definition of regularly varying https://en.wikipedia.org/wiki/Slowly_varying_function.
I'm not sure how you define convergence, and I'm not familiar with SVF, but I tried the expression for small $k$. For $k=1 \ e^{\log a x - \log x} = a$ is a constant, but for $k=2$ $$ e^{\log^2(ax) - \log ^2 x} = e^{\log a \times \log ax^2} = a^{\log a} \times a^{2 \log x} $$ which converges for $0<a \leq 1$ and diverges for $a>1$.