Police are making some ground in the murder mystery of Jennifer Hudson's mother, brother and nephew; but they still aren't calling the person in custody a "suspect." Cops also think they may have found the murder weapon.

A police recruit found a gun in a trash can just a few blocks away from where Hudson's nephew, Julian King, was found dead in the back of an SUV on Monday. Jennifer's mother and brother were, of course, also brutally murdered in their home on Friday.

Authorities are holding King's step-father, William Balfour. He is the estranged husband of Jennifer's sister, Julia. Police say if Balfour did kill the Hudson family, then they suspect he had assistance since he was in custody by Friday evening.

Funeral arrangements are also evidently on hold.