Your best bet would probably be to strip/sand it down to bare wood and
use a slow-set epoxy (not the 5-minute stuff).
If you don't remove the poly, then it doesn't matter how strong the glue
is since the bond between the poly and the wood is likely to be weaker.
The strongest glue I ran across wasn't used as glue but as a two part enamel
paint for an old fiberglass shower tub refinishing job. I had a drop on a
linoleum floor and it couldn't come off without pulling the linoleum apart.
Two part epoxy couldn't stick to the smooth linoleum surface it and could
have come off easily.
Either slow-drying two part epoxy, or polyurethane glue, such as Gorilla.
For epoxy, make sure to mix thoroughly, at least a full minute.
Also, I would sand the poly finish off the lower surface before gluing, at
least rough it up.
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