There is a detailed analysis in Chapters 7 and 8 of Bump and Shilling's Crystal Bases. They work through the connection between RSK and crystals in careful detail, though I don't recall how much detail they give on the variant you are interested in.

There is a detailed analysis in Chapters 7 and 8 of Bump and Schilling's Crystal Bases. They work through the connection between RSK and crystals in careful detail, though I don't recall how much detail they give on the variant you are interested in.