Determine the precise minimax rate

Determine the precise minimax mean-squared-error rate for estimating the constant conditional variance sigma^2 over the considered homoskedastic nonparametric random-design regression classes with regression smoothness beta_b>1 and design-density smoothness beta_g>0.

Background

The paper proves upper bounds for its two-scale estimators and establishes a sharp saturation result for a direct close-pair benchmark, but it does not establish the exact minimax rate for the full model class considered. The authors explicitly identify precise minimax-rate determination as unresolved.