This question came up in a game against Miss Information involving Best of Times Chrono-Ranger.
Miss Information has a "Cat" Stuck in a Tree and an Old Lady in the Streets.
Chrono-ranger has Dead or Alive on "Cat" Stuck in a Tree and The Whole Gang on Old Lady in the Streets.
Chrono-Ranger uses True Purpose, targeting "Cat" Stuck in a Tree.
On Ra's turn, Ra manages to get "Cat" Stuck in a Tree and Old Lady in the Streets to 4hp, then uses his Staff of Ra to destroy "Cat" Stuck in a Tree. Because of True Purpose, The Whole Gang destroys Old Lady in the Streets.
I believe that both bounties would then be destroyed. However, the player playing Chrono-Ranger disagreed, saying that True Purpose prevents the destruction of The Whole Gang. I argued that The Whole Gang doesn't destroy itself when the target of True Purpose leaves play, but will indeed destroy itself when its original target (Old Lady in the Streets) leaves play.
We also argued about whether Dead or Alive (or any bounty for that matter) can be applied to the same target twice using True Purpose, and would also prevent destroying itself if that target leaves play. This is non-trivial, as having those bounties out with no targets would have pretty good benefits pertaining to Chrono-Ranger's equipment.
So, can True Purpose prevent the destruction of bounties in the above fashion?