It's always worked fine for me. You can get a maximum of 6 lives in a duo setup, 5 in solo. 3 base threads, plus 1 bonus per player with the safety net upgrade, plus 1 supplemental thread vision, so yes, it does work with the safety net upgrade(s).
Supplemental thread will appear only for the player with crown (if the group is a duo) and only when you have one thread remaining (so people who want to farm that vision for the achievement can commit suicide until there's just one thread left, and it's possible to get it in Arc 1).