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As Jay serveral times mentioned there is a condition that all odd derivatives are positive. This condition occured also in Szekeres paper [1] however in a slightly different context:
Definition. We call $f(x)$ totally monotonic at $x_0$ if it has derivatives of any order and $(-1)^{k+1}f^{(k)}(x_0)>0$ for every $k>0$.

Then he shows that if the inverse of a function $f$ ($f$ real analytic for $x\ge 0$, $f(x)>x$, $f'(x)>0$ for $x>0$ and $f(x)=x+ax^2+\dots$, $a>0$) is totally monotonic then the regular Abel function is also totally monotonic and is uniquely determined by this property.

$x\mapsto e^x-1$ meets the criteria and its inverse is $x\mapsto \log(x+1)$ and is totally monotonic. Hence the regular Abel function is also totally monotonic.

In our case however the situation is a bit different. The function $e^x$ has no fixed point. $\text{slog}_e$ is an Abel function for it but is not totally monotonic, but the inverse of slog is (/seems to be) totally monotonic.

I would bet there is no proof for the uniqueness claim by total monotonicity, though it sound quite plausible.

[1] G. Szekeres, Fractional iteration of exponentially growing functions, 1961.