I was curious about this issue, so I gave it a try in Poser 12. I used an old Poser character, P6 James.

I made a group on his head, created a prop from it, went into the Hair Room to make some hair, and parented the prop to his head. My "hairwig" was showing as selected, so I went to add it to the Hair library. Got a message telling me to save the hair group with its parent prop, which is what I planned to do anyway. It saved to the Hair library.

New scene (didn't save the James scene or James with the hairwig). Added a new James. Added hairwig to new James. It went right where it was supposed to go and was parented to his head.

Now, I did a crappy job creating this hairwig. Maybe it would have come out differently if I'd used more verts per hair (I knocked it down to 4) or used more hairs (think I used 16,000). I dunno. I don't really use the Hair Room to know what works best, but, in Poser 12, I didn't run into any problems with the prop. Just wanted to make a note of this. Hope your problem can be resolved.

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W11,Intel i9-14900KF @ 3.20GHz, 64.0 GB RAM, 64-bit, GeForce GTX 4070 Ti SUPER, 16GB. 

Old lady hobbyist.

All visual art or fiction is "playing with dolls."