Is it true that when a villager dies before clearing the leaves, the roster of death appears automatically?

nope that isnt true.

Although the roster will not be automatically uncovered if any villagers die before that puzzle is complete, their names will still be recorded and will be listed on the roster once you can view it.

Thank you lady; I have this villager that almost died in an accident at the beginning in my new tribe. I was worried that his name will be lost if he dies before completing the puzzle, and someone told me the opposite. So, naturally I wondered.