Minkowski's ratio-measure power-law convergence conjecture

Let $\mu$ be Minkowski's measure, let $\sigma_j(\mu)$ be the sequence of measures associated with its ratio asymptotics, and let $\sigma_E$ be the expected limiting measure associated with the equilibrium measure on $[0,1]$. Let $d$ denote the Hutchinson distance.

Minkowski's ratio-measure power-law convergence conjecture. There exist two positive constants $A,B$ such that

$$d(\sigma_j(\mu),\sigma_E)<A j^{-B}.$$

The source presents slow numerical convergence of the Hutchinson distance toward zero as evidence for this quantitative limit and for membership of $\mu$ in the Nevai class. No proof is supplied.