Both types of heirloom bounties eventually spawn a unique treasure item (the heirloom) in your room, which you are expected to immediately pick up and store in order to carry back to the guard.
Since the heirloom is dropped in your room, it is possible for another player to pick it up immediately after the heirloom appears. This can be especially problematic for search bounties, since you will always be in roundtime when the heirloom appears.
I think it would be an improvement to make the heirloom spawn in your "at feet" slot rather than generally on the ground. This would admittedly cut down on one kind of player interaction, but to be frank, I don't think it is a very interesting kind.