Existence conjecture for ratios of the third Sierpinski gasket monomial boundary values

Let $r>0$, and let $\gamma_j(r)$ denote the boundary-value sequence associated with the third family of monomials for the $r$-dependent Laplacian.

The ratio-existence conjecture. For all but at most finitely many values of $r$,

$$\lim_{j\to\infty}\frac{\gamma_{j+1}(r)}{\gamma_j(r)}$$

exists. Moreover, there is a continuous function $g:(0,\infty)\to\mathbb{R}$ with $g(r)<0$ for every $r\in(0,\infty)$ such that, whenever the limit exists,

$$\lim_{j\to\infty}\frac{\gamma_{j+1}(r)}{\gamma_j(r)}=g(r).$$

The conjecture proposes a common continuous limiting ratio for the third boundary-value sequence, possibly related to higher Neumann eigenvalues. The paper provides numerical motivation but no proof, so the claim remains open.