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Colour management in this case means just assigning AdobeRGB to the image and converting to sRGB.
It is obvious from the image that it is not only greenish (lack of magenta), but also dull in colour. Usual reason for that is a kind of inconsistancy in colour profiles. From the look of the image (poor reds on the shirt) it seems that it is in AdobeRGB, but rendered as sRGB. I just tagged it correctly by assigning AdobeRGB and then converting to sRGB.
The other case is more complicated, I used there colour management to correct gamma and a little bit - to correct colour.