Convert PNG to GeoTiff using GDAL

I have a map portion PNG and its associated PRJ file. I want to convert the PNG to a GeoTIFF.

I know how to convert a PNG to a GeoTiff if the bounding box is available:

gdal_translate -of Gtiff -a_ullr LEFT_LON UPPER_LAT RIGHT_LON LOWER_LAT -a_srs EPSG_PROJ INPUT_PNG_FILE OUTPUT_GTIFF_FILE. 

But in this case, I don’t have bounding box lat lons specifically. How do I proceed?

My PNG is: NA 10H-N.png and Its gdalinfo says:

Driver: PNG/Portable Network Graphics
Files: NA 10H-N.png
Size is 11575, 4961
Coordinate System is `'
Corner Coordinates:
Upper Left  (    0.0,    0.0)
Lower Left  (    0.0, 4961.0)
Upper Right (11575.0,    0.0)
Lower Right (11575.0, 4961.0)
Center      ( 5787.5, 2480.5)
Band 1 Block=11575x1 Type=Byte, ColorInterp=Palette
  Color Table (RGB with 18 entries...

The associated PRJ file reads;

PROJCS["PROJ",GEOGCS["WGS 84",DATUM["WGS_1984",SPHEROID["WGS 84",6378137,298.257223563,AUTHORITY["EPSG","7030"]],TOWGS84[0,0,0,0,0,0,0],AUTHORITY["EPSG","6326"]],PRIMEM["Greenwich",0,AUTHORITY["EPSG","8901"]],UNIT["degree",0.0174532925199433,AUTHORITY["EPSG","9108"]],AXIS["Lat",NORTH],AXIS["Long",EAST],AUTHORITY["EPSG","4326"]],PROJECTION["Lambert_Conformal_Conic_2SP"],PARAMETER["standard_parallel_1",45],PARAMETER["standard_parallel_2",30],PARAMETER["latitude_of_origin",0],PARAMETER["central_meridian",-80],PARAMETER["false_easting",0],PARAMETER["false_northing",0],UNIT["Meter",1]]