I'm pretty sure "until end of turn" lasts until the next end of turn, not necessarily your end of turn. See Mimicking Miracles and/or Falling Away.
We've mostly replaced "until end of turn" with "this turn".
It's phase specific abilities and counting turns that are specific to a player's turn.
So if Player A activates Jerry's * and player B goes first, souls will be negated for the first two turns?
Seems good.
Based on the above, I think they're just negated for the first turn.