I'm sure if they die without you playing as their household, you lose their personal inventory. If you play as them and they die, their inventory is moved into household inventory.
I lost like a billion pictures because my photographer died without me playing as him, but I had like a billion upgrade parts from my patriarch who scavenged everything since I was playing as his family.
Same thing occurred with my gardening chef; he maintained and harvested his garden often, so he had this practically limitless supply of basil and whatever in his inventory. It was all lost after I found out I left aging on and he died during rotation to a different family. His eldest daughter's inventory which contained a ton of books of life ended up in the household inventory when she died as I was playing with that same family.
So I'm quite sure that things don't end up in the household inventory if you aren't playing as them, including their urns/tombstone.
You can always save before they might die (around times where their bar is all sparkly) to see what happens to the inventory, since I'm not sure if they changed this at all.