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The researcher that authored the paper answered a few questions on Reddit last year [0]. She explained that the color chart isn’t necessary for every photo.

[0] https://www.reddit.com/r/videos/comments/dvts2j/this_researc...

The change in the shades of the color chart (due to lighting) is probably used as a tolerance in the algorithm.

It's necessary for calibration at shooting time. It's a very similar technique to using, say, a ColorChecker Passport or a SpiderCHEKR with their associated profile-creation software for still photography, the main difference being that attenuation underwater is much more variable that lighting conditions on dry land. You need to accommodate attenuation of both the incident light (light hitting the subject has to get there through local water conditions) and light reflected from the subject at different depths at different times of the day and year, and with different "water" compositions (minerals, particulates, etc.).