Before being inducted into MiB, K takes J into Jeebs' Pawn Shop. Jeebs calls him "Officer Edwards" (Knowing J's name)
Jeebs obviously knows J from his pre-MiB life. When J's history is erased do they do so with the knowledge that Jeebs knows his identity?
The neutralizer can wipe backwards but doesn't seem to have selective memory removal. (To get rid of J's life in Jeebs head would require blanking Jeebs to before he ever met J)