Conjecture on asymptotic growth of cluster scattering wall-function coefficients

Let $b,c>0$ with $bc>4$, and let $r$ lie in the Badlands interval

$$r\in\left(\frac{bc-\sqrt{bc(bc-4)}}{2c},\frac{bc+\sqrt{bc(bc-4)}}{2c}\right).$$

For positive integers $i,j$, let $\tau(i,j)$ be the corresponding wall-function coefficient. Asymptotic growth conjecture. For every such $r$, the limit

$$f(r)=\lim_{j/i\to r,\ i\to\infty}\frac{\ln\tau(i,j)}{\sqrt{bcij-bi^2-cj^2}}$$

exists and is continuous as a function of $r$. This predicts a uniform asymptotic growth law for wall-function coefficients in the Badlands. The wall-functions are known to be algebraic in the central-wall situation, but the analogous algebraicity and general exponential-growth behavior remain open questions; the stated limit is proposed on the basis of numerical experiments.