Minkowski's orthogonal-polynomial zero convergence conjecture

Let $\mu$ be Minkowski's measure, and let $\zeta_l^j$, $l=1,\ldots,j$, be the zeros of its degree-$j$ orthogonal polynomial. Let $\theta_l^j$ be the zeros of the degree-$j$ Chebyshev polynomial on $[0,1]$, given by

$$\theta_l^j=\frac12\left[1-\cos\left(\frac{2l-1}{2j}\pi\right)\right],\qquad l=1,\ldots,j.$$

Minkowski's orthogonal-polynomial zero convergence conjecture. As $j$ tends to infinity, the zeros $\zeta_l^j$ converge uniformly to the zeros $\theta_l^j$; moreover, there exist positive constants $A,B$ such that

$$U_j=\max_{1\leq l\leq j}|\theta_l^j-\zeta_l^j|\leq A j^{-B}.$$

The claim would imply convergence of the zero distributions to the equilibrium measure on $[0,1]$ and hence support regularity. The source gives numerical support but no proof.