Starter thread, at least back in time (2022) it was not easy/possible to use ffmpeg for adding custom icc profile to mp4 container but MP4Box from gpac project was up to task:

https://superuser.com/questions/1739082/embedding-a-custom-icc-profile-in-an-mp4-with-ffmpeg

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Creating MP4 video without ICC profile (for testing):
ffmpeg -y -loop 1 -framerate 1 -i icc_chelsea.png -vcodec libx265 -t 10 chelsea.mp4

Extracting the ICC profile:
magick icc_chelsea.png profile.icc

Adding the ICC profile to the MP4 video file using MP4Box:
MP4Box -add video.mp4#video:colr=prof,profile.icc -new icc_video.mp4

Testing with FFprobe:
ffprobe icc_chelsea.mp4

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As far as I understand in this case idea was to embed display profile into media. But few apps outside of Final Cut Pro/quicktime can do anything useful about this info?

https://forum.luminous-landscape.com/index.php?topic=129232.0

this one from ~2019.

It states that instead of more flexible color management video world used few fixed mode monitor color spaces, and whole process was a bit manual if your input image/video was not in one of those.

I think we hit crudest form of this limitation when we try to fade titler's text over png image. It sort of flashes, due to missing (unapplied) 2.4 gamma step?