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The PRad experiment has credibly demonstrated the advantages of the calorimetric method in e-p scattering experiments to measure the proton root-mean-square (RMS) charge radius with high accuracy. The PRad result, within its experimental uncertainties, is in agreement with the small radius measured in muonic hydrogen spectroscopy experiments and it was a critical input in the recent revision of the CODATA recommendation for the proton charge radius. Consequently, the PRad result is in direct conflict with all modern electron scattering experiments. Most importantly, it is 5.8% smaller than the value from the most precise electron scattering experiment to date, and this difference is about three standard deviations given the precision of the PRad experiment. As the first experiment of its kind, PRad did not reach the highest precision allowed by the calorimetric technique. Here we propose a new (and) upgraded experiment -- PRad-II, which will reduce the overall experimental uncertainties by a factor of 3.8 compared to PRad and address this as yet unsettled controversy in subatomic physics. In addition, PRad-II will be the first lepton scattering experiment to reach the Q^2 range of 10^{-5} GeV^2 allowing a more accurate and robust extraction of the proton charge radius. The muonic hydrogen result with its unprecedented precision (~0.05%) determines the CODATA value of the proton charge radius, hence, it is critical to evaluate possible systematic uncertainties of those experiments, such as the laser frequency calibration that was raised in recent review articles. The PRad-II experiment with its projected total uncertainty of 0.43% could demonstrate whether there is any systematic difference between $e-p$ scattering and muonic hydrogen results. PRad-II will establish a new precision frontier in electron scattering and open doors for future physics opportunities.