it only affects units in the black heart detachment.
If that unit has power from pain, kabalites or archon, they treat it as 1 round higher.
If the unit does not have power from pain, ravager or razorwing jet-fighter, it gets the benefits from round 1 inured to suffering and that's it. 6+++ feel no pain save.
Mercenaries, scourge or mandrakes, never get obsession benefits. If their data card says power from pain they use the current round number to determine their benefits, they do not get the black heart buff.
Bear me thither so that my blade might taste of their flesh.