A conjectured uniqueness criteria for analytic tetration

[JmsNxn]

So I feel like the solution to tetration whose derivative is completely monotone is definitely unique. In attempts at solving this, the biggest obstacle I found, one I avoided and just assumed, is that the exponential function is the only completely monotone solution to some multiplicative equations. I posted the question on MO
http://mathoverflow.net/questions/260298...-equations
I think if we have this complete monotonicity will follow from this. This is mostly because of Sheldon's proof that the schroder tetration is completely monotone.