Asymptotic growth conjecture for the balanced embedding solution

Let $f^\beta(x)$ be the unique solution of equation (2) with $f^\beta(0)=1$. Asymptotic growth conjecture. As $x\to+\infty$, the function $f^\beta$ should be asymptotic to

$$C_\beta x^\beta e^x$$

for a constant $C_\beta$. The conjecture concerns the asymptotic behaviour of the model solution at infinity; the preceding result establishes uniqueness and continuous dependence on $\beta$, while the stated asymptotic remains to be proved.