Chihara's conjecture on the essential spectrum of Jacobi matrices

Let $C$ be a self-adjoint Jacobi matrix, let $b_n\to\infty$, and suppose that the smallest point $\rho$ of $\sigma_{ess}(C)$ is finite. Assume

$$\lim_{n\to\infty}\frac{a_n^2}{b_n b_{n+1}}=\frac14.$$

Chihara's conjecture. The essential spectrum satisfies

$$\sigma_{ess}(C)=[\rho,\infty).$$

Chihara proved under these assumptions that the zeros of the associated orthogonal polynomials are dense in $[\rho,\infty)$; the conjecture asks for the corresponding full description of the essential spectrum.