Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). $$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). $$
This equation has many solutions.
Compute a solution to that equation with the fixed point at $0$ and its Koenigs function, and call the solution $k(x)$.
So
$$k(2x) = A(k(x)). $$
Then it seems it is always true that
Conjecture $T$:
$$G(x) = k(x)$$ $$ G(x+y) = k(x+y) = f(k(x),k(y)) $$
Of course we can not use The Koenigs function if its conditions are not met, and there is no way around it. In other words The fixed point of $A(x)$ (at $0$) should not be parabolic and must be strictly positive.
I searched the internet for Koenigs function and formal group law but did not find them combined.
Is conjecture $T$ true? How is it proved?