The K7 T Tauri star PDS 70 remains the best laboratory for investigating the influence of giant planet formation on the structure of the parental disk. One of the most intriguing discoveries is the detection of a resolved inner disk from ALMA observations that extends up to the orbit of PDS 70b. It is challenging to explain this inner disk because most of the dust particles are expected to be trapped at the outer edge of the gap opened by PDS 70b and PDS 70c. By performing dust evolution models in combination with radiative transfer simulations that match the gas disk masses obtained from recent thermo-chemical models of PDS 70, we find that when the minimum grain size in the models is larger than 0.1 µm, there is an efficient filtration of dust particles, and the inner disk is depleted during the first million year of dust evolution. To maintain an inner disk, the minimum grain size in the models therefore needs to be smaller than 0.1 µm. Only when grains are that small are they diffused and dragged along with the gas throughout the gap opened by the planets. The small grains transported in the inner disk grow and drift into it, but the constant reservoir of dust particles that are trapped at the outer edge of the gap and that continuously fragment allows the inner disk to refill on million-year timescales. Our flux predictions at millimeter wavelength of these models agree with ALMA observations. These models predict a spectral index of 3.2 in the outer and 3.6 in the inner disk. Our simple analytical calculations show that the water emission in the inner disk that was recently observed with the James Webb Space Telescope may originate from these ice-coated small grains that flow through the gap, grow, and drift toward the innermost disk regions to reach the water snowline. These models may mirror the history and evolution of our Solar System, in which Jupiter and Saturn played a crucial role in shaping the architecture and properties of the planets.