gdalwarp - GDAL and reprojecting a NetCDF file

I have a .nc file that you can download here. This is satellite imagery.

I want to re-project this and save it to a new .nc file, without saving to raster. Just NetCDF to NetCDF only. Is this possible?

I am getting an error saying "ERROR 4: `test.nc' not recognized as a supported file format."

The code I have tried is :

gdalwarp -t_srs EPSG:3857 test.nc test-projected.nc

I am not a professional with this, but should I even be using gdalwarp to do this? (maybe gdaltransform?) I only want to change the projection - nothing else at this time.