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Forward proton spectrometers at high-energy proton colliders rely on precision timing to discriminate signal from background. Silicon low gain avalanche diodes (LGADs) are a candidate for future timing detectors in these systems. A major challenge for the use of LGADs is that these detectors must be placed within a few mm of the beams, resulting in a very large and highly non-uniform radiation environment. We present a first measurement of the current and capacitance vs. voltage behavior of LGAD sensors, after a highly non-uniform irradiation with beams of 24 GeV/c protons at fluences up to $1\times10^{16} p/cm^{2}$.