Looks like there is a lot of support for Old Brown Glue. It certainly will be an
easier option than full disassembly, which I likely would need to do if I went
to traditional hide glue, while keeping the main benefits.

FWIW: going with what I am comfortable with here would presume I have more
experience with hide and fish glues than I do. My go to has been modern
adhesives for things that don't need reversibility or historical correctness.
This case (pun intended) deserves both.

Thank you all for the input. I might try Tom's water-and-warm suggestion, first.
No clue if it will work, sorta work, or not work, but might be interesting to
find out.

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