For 3N + 1GF, I would guess 3 kills and 1 investigation.

The first geisha died too quickly to have been researched first. They got lucky. Gregoshi was trying to play a nice balance between over-active and "I'm new," which I suspect delayed him getting whacked at first but did NOT stop them from investigating. Once known.....

That is all supposition of course, but I think it fits the facts and the kill descriptions thus far.