Asymptotic ratio conjecture for the second Sierpinski gasket monomial boundary values

Let $r>0$, and let $\beta_j(r)$ denote the boundary-value sequence associated with the second family of monomials for the $r$-dependent Laplacian. Let $\lambda_2(r)$ and $\lambda_3(r)$ be the corresponding Neumann eigenvalues.

The asymptotic ratio conjecture. For $r\neq \frac{\sqrt{17}-3}{4}$,

$$\lim_{j\to\infty}\frac{\beta_{j+1}(r)}{\beta_j(r)}=\frac{-1}{2\lambda_3(r)-\lambda_2(r)}.$$

The claim is supported by the paper's numerical observations and by the known behavior at $r=1$, but the text notes visible finite-$j$ discrepancies and does not prove the general limit. It therefore remains open.