Universal uniqueness criterion?

[BenStandeven]

What about using Kneser's approach to produce the alternate tetration functions?

There is no initial region connecting an alternative fixed point pair.
The image of the straight line connecting a secondary fixed point pair overlaps with itself.
Secondary fixed points lie in a range with imaginary part greater or less than pi.
The vertical line connecting a pair is longer than 2*pi.
This means the image revolves more than once around 0 with constant radius, hence overlapping itself.

I tried to construct different connecting lines of a secondary fixed point pair and failed. I believe there is no initial region connecting two secondary fixed points.

Yeah, that's right; the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point.