Using a large dataset on major stock indexes and FX rates, we test the robustness of the rough fractional volatility model over different time scales. We include the estimation error as well as the microstructure noise into the analysis. Our findings lead to new stylized facts regarding the volatility that are not described by models introduced so far: in the fractal analysis using the absolute moment approach, log-log plots are nonlinear and reveal very low perceived Hurst exponents at small scales, consistent with the rough framework, and higher perceived Hurst exponents for larger scales, along with stationarity of the volatility. These results, obtained for time series of realized volatilities are confirmed by another measure of volatility, namely Parkinson's volatility, taking into account its specificities regarding measurement errors.