Manirul Islam, an MLA of the ruling Trinamool Congress, is the prime accused in the murders.
The plea was filed by the victims' family before the court of Justice Dipankar Dutta, seeking a probe by the Central Bureau of Investigation (CBI) into the 2010 killings of three brothers in Labhpur of the district.
While Labhpur legislator Islam was named in the first information report (FIR) filed after the killings, his name was subsequently dropped from the charge sheet filed four years later.
Incidentally, Islam in July 2013 publicly admitted to committing the murders.
"I have crushed three people to death with my feet," Islam boasted in his address at a party rally that was video-graphed and widely broadcast and reported upon.
"Justice Dutta Monday watched the video in detail, and today (Tuesday) we submitted a transcript of Islam's speech in the video. The court has directed the state government to state its stand about the petition in an affidavit by Aug 1," said Subrata Mukherjee, counsel for the petitioner.