We can query the image compression e.g. via gdalinfo img.tif which gives us (among other information) something like

Image Structure Metadata:
  COMPRESSION=DEFLATE
  INTERLEAVE=BAND
  LAYOUT=COG

But the image compression type can be different for the overviews (by default it is the same for image and overviews) and I am wondering if there is any way to query what the overview compression is?