The asymptotic growth conjecture for Golomb-like systems

Let $c_f$, $d_f$, $c_g$, and $d_g$ be integers with $d_f+d_g>0$, and consider the Golomb-like system

$$\begin{cases} f(n)=g(n-g(n-1)-c_f)+d_f\\ g(n)=f(n-f(n)-c_g)+d_g. \end{cases}$$

Asymptotic growth conjecture. Any solution to this system grows asymptotically like

$$\sqrt{\left(d_f+d_g\right)n}.$$

This conjecture proposes a common asymptotic growth rate for all solutions of these nested recurrences, generalizing the analogous conjectured behavior of Golomb's recurrence. The source provides no evidence that it has been resolved.