Exponential form of the bounding functions for Goldbach partitions

Let $G(n)$ denote the number of pairs of primes whose sum is the even integer $n$. Let $f_L(x)$ and $f_U(x)$ be monotonous analytic lower and upper bounding functions satisfying

$$f_L(n)<G(n)<f_U(n)$$

for every even integer $n\geq 6$. Numerical computations suggest that these functions satisfy a scaling functional equation, and the relevant exponential form has constants $\alpha>0$ and $0<\beta<1$.

Exponential bounding-functions conjecture. The lower and upper bounding functions of $G(n)$ can be expressed respectively in the form

$$f_L(x)=\exp(\alpha_L x^{\beta_L}),\qquad f_U(x)=\exp(\alpha_U x^{\beta_U}),$$

where the constants satisfy $\alpha_L,\alpha_U>0$ and $0<\beta_L,\beta_U<1$ and can be determined by numerical computations.

This conjecture is motivated by numerical computations for $n<1{,}000{,}000$ and a functional equation for the normalized bounding functions. The paper gives no proof or resolution.