Audio Bleed From Too Many Mics/Tracks

I'm editing a multi channel game show video that has 7 tracks of audio. Each track is from a different person who was mic'd up with a lav. They were all standing close to each other in the studio, and I'm getting audio bleed from everyone talking. In other words, Person A's mic picked up Person B talking next to them and so on.

Is there a plugin or method in Premiere or Audition that can target the primary speaker's audio and drown out the person(s) talking next to them? 

I'm edting the videoin Premiere on an iMac.