On measures generating orthogonal polynomials with similar asymptotic behavior of the ratio at infinity

We consider the influence of the measure perturbations on the asymptotic behavior of the ratio of orthogonal polynomials. We suppose that the absolutely continuous part of the measure is supported on finitely many Jordan curves. The weight function satisfies the modified Szegö condition.
The singular part of the measure consists of finitely many point masses outside the polynomial convex hull of the support of the absolutely continuous part of the measure. We study the stability of asymptotics of the ratio in the following sense:
$$ \frac{P_{\nu,n}(z)}{P_{\nu,n+1}(z)}-\frac{P_{\mu,n}(z)}{P_{\mu,n+1}(z)}\to 0,\quad n\to\infty. $$
The problem is a generalization of the problem on compactness of the perturbation of Jacobi operator generated by the perturbation of its spectral measure. We find a condition necessary (or necessary and sufficient under some additional restriction) for the stability of the asymptotical behavior of the corresponding orthogonal polynomials. One of the main tools in the study are the Riemann theta functions.