Your assumption is right, you need the correct values for the corner coordinates.

Since the source image is slightly larger than the Earth, you can not simply reproject the corner coordinates. You may use a different approach:

The data is 5500x5500pix, providing a 2km resolution on the center. So the extent from the center is 5500/2*2000m = +/-5500000m.

With that, and the WGS84 ellipsoid data, I used this transformation:

gdal_translate -a_srs "+proj=geos +h=35785863 +a=6378137.0 +b=6356752.3 +lon_0=140.7 +no_defs" -a_ullr -5500000 5500000 5500000 -5500000 PI_H08_20150125_0230_TRC_FLDK_R10_PGPFD.png temp.tif
gdalwarp -overwrite -t_srs "+proj=latlong +ellps=WGS84 +pm=140.7" -wo SOURCE_EXTRA=100 temp.tif Himawari8.tif

Your assumption is right, you need the correct values for the corner coordinates.

Since the source image is slightly larger than the Earth, you can not simply reproject the corner coordinates. You may use a different approach:

According to the project site, the 5500x5500pix data is providing a 2km resolution on the center. So the extent from the center is 5500/2*2000m = +/-5500000m.

With that, and the WGS84 ellipsoid data, I used this transformation:

gdal_translate -a_srs "+proj=geos +h=35785863 +a=6378137.0 +b=6356752.3 +lon_0=140.7 +no_defs" -a_ullr -5500000 5500000 5500000 -5500000 PI_H08_20150125_0230_TRC_FLDK_R10_PGPFD.png temp.tif
gdalwarp -overwrite -t_srs "+proj=latlong +ellps=WGS84 +pm=140.7" -wo SOURCE_EXTRA=100 temp.tif Himawari8.tif