Just did some further testing on Acrobat X...
This appears to a "standard everyday colour management issue" with the difference from Adobe RGB and sRGB...
Testing with both device RGB and ICC RGB...
Acrobat colour settings for a device RGB set to sRGB show this colour issue using convert colours, which is expected.
However, even when the source image was tagged as Adobe RGB, it appears that convert colour was still using colour settings and not the embedded profile as the source.
Only when I changed colour settings to Adobe RGB did the image tagged with Adobe RGB convert correctly.
Could it be that Convert Colours ignores embedded profiles and simply using colour settings RGB instead? This was a test of a single image, if there were multiple images it would be even worse as they could all have different ICC tags assigned to them.
EDIT: Acrobat Pro DC appears to behave the same, object-level tag ignored as source profile, colour settings profile used as the source.