Does killing wanted ships still lower influence

Does killing wanted ships of the controlling faction still lower their influence or do you have to target clean ships? I'm just doing some bounty hunting but I don't want the factions influence to go down, I'm only killing wanted ones.
 
I understand that the influence might have a trivial decrease when you kill a faction's wanted ships. But if the faction is the controlling faction you get a significant increase in influence when you turn in the bounties in that system.

Watch out with killing clean ships. That can lead to lock down.
 
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