Conjecture on zero spacing near an algebraic singularity for regular measures

Let $\mu$ be a finite Borel measure supported on a compact subset $K$ of the real line and assume that $\mu$ is regular in the sense of Stahl and Totik on $K$. Let $x_0\in\operatorname{int}(K)$, and suppose that $\mu$ is absolutely continuous in a neighbourhood $(x_0-\varepsilon,x_0+\varepsilon)$ of $x_0$, with

$$d\mu(x)=w(x)|x-x_0|^\lambda dx,\qquad x\in(x_0-\varepsilon,x_0+\varepsilon),$$

where $\lambda>-1$ and $w(x)$ is strictly positive and continuous. Let $\nu_K$ denote the equilibrium measure of $K$, let $\omega_K$ denote its equilibrium density, and let $j_k(a,c,d)$ denote the zeros of the referenced Bessel equation in the specified order. Zero-spacing conjecture. For fixed $k\in\mathbb Z$, if $\nu_K([x_0,\infty))=p/q$ is rational with $\gcd(p,q)=1$, then there are $q$ distinct constants $c_0,\dots,c_{q-1}\in\mathbb R$ and $q$ distinct constants $d_0,\dots,d_{q-1}\in\mathbb R$ such that, for $n_l=lq+m$, $m=0,1,\dots,q-1$,

$$\lim_{l\to\infty}n_l\pi\omega_K(x_0)\bigl(x_{k+1,n_l}(x_0)-x_{k,n_l}(x_0)\bigr)=j_{k+1}\left(\frac{\lambda-1}{2},c_m,d_m\right)-j_k\left(\frac{\lambda-1}{2},c_m,d_m\right).$$

Moreover, the constants $c_l,d_l$ for two distinct $k_0,k_1\in\mathbb Z$ are equal if $k_0$ and $k_1$ have the same sign. If $\nu_K([x_0,\infty))$ is irrational, then for any $c,d\in\mathbb R$ there is a subsequence $n_l$ such that

$$\lim_{l\to\infty}n_l\pi\omega_K(x_0)\bigl(x_{k+1,n_l}(x_0)-x_{k,n_l}(x_0)\bigr)=j_{k+1}\left(\frac{\lambda-1}{2},c,d\right)-j_k\left(\frac{\lambda-1}{2},c,d\right).$$

The conjecture extends the paper's explicit zero-spacing result from measures on $[-1,1]$ with a prescribed algebraic singularity to general regular measures. It predicts that the limiting fine spacing is governed by the equilibrium density and by rational or irrational arithmetic of the equilibrium mass to the right of the singularity; the general case remains open.