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Zonal statistics with GDAL only (not Python)

This is not a command-line solution however given the nature of need in the question, I think it might help be helpful. If my understanding is correct, you want to get the summary statistics for the ...
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Resampling high resolution data into lower resolution based on most common value

You should be able to do this by resampling the 3cm raster to 20m using the "majority" method in ArcGIS Resample. See
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