Agree with @BobKamman . Only the parent who the kid lived with is eligible to file HOH. The other must file MFS. 
New Jersey does not recognize a separation agreement notarized or not. They must file MFS or MFJ the taxpayer would have to file MFS. The parent with the kids in residence could file HOH

They would be better off filing MFJ.

At @BobKamman  seriously? I’ll be back in a little bit to tell you about New Jersey fuel.