Zero-spacing conjecture for orthogonal polynomials on Cantor-type sets

For a measure $\mu$ supported on $\mathbb{R}$, let $Z_n(\mu)$ be the zero set of its degree-$n$ orthogonal polynomial. For $n>1$, define

$$M_n(\mu):=\inf_{\substack{x,x'\in Z_n(\mu)\x\neq x'}}|x-x'|.$$

Enumerate $Z_N(\mu_{K(\gamma)})$ as $x_{1,N}<\cdots<x_{N,N}$ and set

$$A_{n,N}:=|x_{2n,N}-x_{2n-1,N}|, \qquad A_N:=\max_{n,m\in\{1,\ldots,N/2\}}\frac{A_{n,N}}{A_{m,N}}.$$

Zero-spacing conjecture. For each $\gamma=(\gamma_k)_{k=1}^\infty$ with $\inf_k\gamma_k>0$, $(A_{2^k})_{k=1}^\infty$ is unbounded. If $s=2^k$ for some $k\in\mathbb{N}$, there is a $c_0\in\mathbb{R}$ depending on $k$ such that

$$\lim_{n\rightarrow\infty}\frac{A_{s,2^n}}{A_{1,2^n}}=c_0.$$

The conjecture describes global growth and selected limiting ratios of zero spacings; the source supports it with numerical experiments and excludes small-$\gamma$ regimes where boundedness is known. Its resolution is not given.