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An exomoon will produce transit timing variations (TTVs) upon the parent planet and their undersampled nature causes half of such TTVs to manifest within a frequency range of 2 to 4 cycles, irrespective of exomoon demographics. Here, we search through published Kepler TTV data for such signals, applying a battery of significance and robustness checks, plus independent light curve analyses for candidate signals. Using the original transit times, we identify 11 (ostensibly) single-planets with a robust, significant and fast ($P_{TTV}<4$ cycles) TTV signal. However, of these, only 5 are recovered in an independent analysis of the original photometry, underscoring the importance of such checks. The surviving signals are subjected to an additional trifecta of statistical tests to ensure signal significance, predictive capability and consistency with an exomoon. KOI-3678.01, previously validated as Kepler-1513b, is the only case that passes every test, exhibiting a highly significant (>20 sigma) TTV signal with a periodicity, amplitude and shape consistent with that caused by an exomoon. Our analysis finds that this planet is $8.2_{-0.5}^{+0.7}$ $R_{\oplus}$ orbiting at $0.53_{-0.03}^{+0.04}$ AU around a late G-type dwarf. After forecasting the planetary mass, we expect it to be capable of maintaining at least a 0.3 $M_{\oplus}$ exomoon for 5 Gyr, and the TTV signal corresponds to a moon mass as low as 0.75 Lunar masses. We thus encourage follow-up observations and dynamical analysis of this unique signal, but caution skepticism until such data can be obtained.