Asymptotic ratio conjecture for the first Sierpinski gasket monomial boundary values

Let $r>0$, and let $P_{j,1}^{(r)}$ and $P_{j,2}^{(r)}$ be the first two families of monomials associated with the $r$-dependent Laplacian. Write their relevant boundary values as $\alpha_j(r)$ and $\beta_j(r)$, respectively, and let $\lambda_2(r)$ and $\lambda_3(r)$ denote the corresponding Neumann eigenvalues.

The asymptotic ratio conjecture. For $r\neq 1$ and $r\neq \frac{\sqrt{17}-3}{4}$,

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

This predicts the exponential decay ratio of the first boundary-value sequence and is motivated by the numerical data on monomials and the spectrum of the $r$-dependent Laplacian. The source does not establish the limit, so its resolution remains open.