The statement below is false.
The statement above is correct.
There is a tie breaker:
Note: If both Captains lose their last “Crew” the same round, the battle is won by the Captain who made the most successes. Resolve ties by adding up numbers on the dice that did not come up as successes. If still tied, combat ends with no winner.
I'd have to guess what happens if the tie breaker ties as well. My guess
would be a) he doesn't die, and b) the card doesn't get discarded, but that's really just a guess (edit) was at least half wrong.