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I've had problems with Liquid Nails and styrofoam - it stops sticking after a while and comes apart. So I don't use it anymore and stick to PVA (sorry, bad pun).
That is interesting. I have had a different experience with Liquid Nails. I can't get the (expletive deleted) stuff off plywood.The Styrofoam is not strong enough to resist much force so it breaks off the glue. The glue however is very difficult to remove from salvaged plywood. At least you can sand PVA down if the Styrofoam breaks off the glue.
If you properly clamp and
press Styrofoam to plywood with PVA or Liquid Nails you will not get it off in one piece.
I think the PVA ( a carpenters brand not school glue) would create a stronger bond if it is
pressed and clamped because the PVA will find its way into the porous Styrofoam and grip better. The Styrofoam products sold for home insulation have a vapor barrier that might prevent PVA from bonding at all.Remove it and ruff up the surface before you glue and clamp. It will take a week to cure. If it is a small piece It might be
ok after a few days.