Universal uniqueness criterion?

[Kouznetsov]

*What is the problem with the extension overlapping with the original domain?
*What initial regions do all of the known methods of complex extensions of tetrations have? If their initial regions don't deviate too much from each other that their exponentials leave the domain of biholomorphy, we can prove the weaker theorem that all of the known extensions are the same. Can anyone show the (mapping graphs on the complex plane) exponentials and the logarithms of the intiial regions?

The region G where the tetrtion is biholomorphic is plotted in figure 2 of
http://www.ils.uec.ac.jp/~dima/PAPERS/2009uniabel.pdf
exp(G) is shown at the right (between green and blue curves) and
log(G) is shown at the legt (between the red curve and the black line).
One has no need to evaluate tetration in order to plot this figure.
The region G is filled with lines of constant real part of arctetration and lines of constant imaginary part in figure 1b of
http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf
You may extract the figures mentioned and just overlap them.