Note that if this happened before the objective was revealed, the player would come in with a new investigator...
Certain Mythos cards in the Green-Eyed Boy can make that not happen. IIRC, you don't get to come in with a new investigator if you die while under the effects of Loss of Will or Selfish Motives even if the objective has not been revealed. I think one of the CotW trauma cards provides another way to die without getting a new investigator even with an unrevealed objective.
The way I read the OP, it seemed to imply that the killed investigator had either Loss of Will or Selfishly Motivated. If that is not the case, and the objective was unrevealed, then yes, the killed investigator should be replaced next game round.