Transmitter channel state information (CSIT) is indispensable for the spectral efficiency gains offered by massive multiple-input multiple-output (MIMO) systems. In a frequency-division-duplexing (FDD) massive MIMO system, CSIT is typically acquired through downlink channel estimation and user feedback, but as the number of antennas increases, the overhead for CSI training and feedback per user grows, leading to a decrease in spectral efficiency. In this paper, we show that, using uplink pilots in FDD, the downlink sum spectral efficiency gain with perfect downlink CSIT is achievable when the number of antennas at a base station is infinite under some mild channel conditions. The key idea showing our result is the mean squared error-optimal downlink channel reconstruction method using uplink pilots, which exploits the geometry reciprocity of uplink and downlink channels. We also present a robust downlink precoding method harnessing the reconstructed channel with the error covariance matrix. Our system-level simulations show that our proposed precoding method can attain comparable sum spectral efficiency to zero-forcing precoding with perfect downlink CSIT, without CSI training and feedback.