Before revision:   We also demonstrate in  simulation study that our procedure has good small sample properties.
After revision:  To construct simultaneous confidence regions on many target parameters, sufficiently fast estimation rates of the nuisance functions are crucial. In this context, we establish uniform estimation rates and sparsity guarantees of the square-root estimator in a random design under approximate sparsity conditions that might be of independent interest for related problems in high-dimensions. We also demonstrate in simulation study that our procedure has good small sample properties.