White plastazote foam.
Plastazote is a soft, versatile, closed cell, cross-linked expanded polyethylene foam. Application areas include prop making, cosplay, costume design, LARP, packaging, protective padding in contact sports, automotive, health care and building.
Our supplier tells us that this is used by museums for storing items when not on display.
Plastazote blunts scalpel blades quite quickly, so change the blade if the plastazote starts to bobble.
Stick with a contact adhesive, practice gluing plastazote with some scrap material.
With a spreading stick, spread a layer as thin as possible completely covering both surfaces, let the glue become touch dry, then press the plastazote pieces together firmly for at least 10 seconds.
Be accurate as the glue is not repositionable.