The multiplicity-growth limit conjecture for automata-group transition operators

Let $M_{k=j}^{(n)}$ be the transition operator indexed by the non-negative integer $j$, let $\operatorname{Sp}(M_{k=j}^{(n)})$ denote its spectrum, and let $m^{(n;j)}(\lambda)$ be the multiplicity of the eigenvalue $\lambda$ in $M_{k=j}^{(n)}$. Multiplicity-growth limit conjecture. For every non-negative integer $j$, there exist integers $n_{\lambda,j}\in\mathbb{N}$ such that

$$\lim_{n\to\infty}\operatorname{Sp}\left(M_{k=1}^{(n)}\right)=\lim_{n\to\infty}\left\{\lambda\in\operatorname{Sp}\left(M_{k=j}^{(n)}\right)\,\middle|\,0<m^{(n_{\lambda,j};j)}(\lambda)\leq\cdots\leq m^{(n-1;j)}(\lambda)\leq m^{(n;j)}(\lambda)\right\}.$$

The conjecture asserts that the limiting spectrum for $k=1$ agrees with the limiting collection of eigenvalues whose multiplicities are positive from some index onward and nondecreasing through the stages displayed. It is motivated by the reported spectral distributions and multiplicity-growth observations, but no proof or resolution is supplied in the source.