Ignjatović's asymptotic Christoffel-Darboux kernel conjecture

Let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ be the recurrence coefficients, and let $p_j(x)$ denote the corresponding orthonormal polynomials. Write $\Delta a_n=a_{n+1}-a_n$ and $\Delta^2a_n=\Delta(\Delta a_n)$. Assume:

• $\lim_{n\to\infty}a_n=\infty$;
• $\lim_{n\to\infty}\Delta a_n=0$;
• there exist $n_0,m_0$ such that $a_{n+m}>a_n$ for all $n\geq n_0$ and all $m\geq m_0$;
• $\sum_{n=0}^{\infty}1/a_n=\infty$;
• there exists $\kappa>1$ such that $\sum_{n=0}^{\infty}1/a_n^\kappa<\infty$;
• $\sum_{n=0}^{\infty}|\Delta a_n|/a_n^2<\infty$;
• $\sum_{n=0}^{\infty}|\Delta^2a_n|/a_n<\infty$.

Ignjatović's conjecture. If

$$-2<\lim_{n\to\infty}\frac{b_n}{a_n}<2,$$

then, for every $x\in\mathbb{R}$, the limit

$$\lim_{n\to\infty}\left(\sum_{j=0}^n\frac{1}{a_j}\right)^{-1}\sum_{j=0}^np_j^2(x)$$

exists and is positive.

The conjecture predicts a positive normalized limiting density for the Christoffel-Darboux kernel under the stated growth, difference, and summability conditions on the recurrence coefficients. The supplied source does not indicate whether this conjecture has been resolved.