Fatchoppers,
This is what I would do,
That area isn`t too big, I would wet a sponge and squeeze out the water till it`s almost damp. Place the sponge on the area, or sponges and let them sit there for about 4 hours. (wet the tops if nessecarry)
After removeing the sponges, take Elmer`s water based wood glue (shake the glue well) and make a line on the wet cracking. Then take an old toothbrush and work the glue under the cracks. take your time and used a few bristles to get in there.
Then take a new piece of Aluminum foil (it can wrinkle a little bit) and cover the wet area`s while the glue is still wet. Now you can use a thin piece of wood or a thin piece of plastic that covers the areas. Use a bunch of clamps to fourm the thin piece over the area flat because it`s curved. PROTECT THE OTHER SIDE TOO !!
After it dries for at least 24HR., more is better, you can remove the clamps and thin piece, then the foil (some can stick) and use a kitchen sponge scrub side , dampened, to rub off the foil and extra glue.
All this may seem complicated but it`s not,...it`s quicker than most think to do. The idea is to soften the layer that`s lifting and use clamps to close the seams so to speak.