An idea I had was to measure the length from the brow right above the eyes to the back edge.
Then cut wood to that length plus some more, and at the ends make it angled.
Attach magnets to the end of the wood and in the bio where the wood meets it.
Then the other end that is angled mounts to a rectangle of wood that has a slot in it at the top edge, it gets glued.
Paint the wood the color of the bio.
Then that assembly can be hung on a peg or nail.
Here's a concept drawing, the side view shows dark rectangles at the end of the wood piece- these are the magnets.
The front view shows what it looks like when there is no bio mounted:
The angle where the wood piece meets the rectangle of wood could be adjusted so the back jaw part of the bio rests on the wall.
There could be more magnets- at the back end of the bio and that spot on the wood also, for security.
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Ei'luj