I tried compressing BC7 (from AMD-s Compressonator) using lz4c and it wasn't much.
Just re-ran it (with -hc, version 1.9.3(latest now)) and: "Compressed 5592544 bytes into 5413783 bytes ==> 96.80%".
7z with "7z a -mx=9 -t7z asd.7z albedo_PNG_BC7_1.KTX" does 5592544 bytes to 4844707 bytes (down to ~87%).
Original file is 10227047 bytes (PNG, RGBA), i can't remember if the ktx has mipmaps.
EDIT: Note that the image is fairly noisy (gravel). Could/should be better with more "artificial" textures.
I don't know if ktx does some extra compression, but, looking at it, i doubt it.
PS I think that BC could be massaged at compression to be better compressible, and i think i read something about that. Don't remember.