I assume you redrew the drawing to include some wiring- I can't get it either.
I'll assume that the hall lights are controlled by two of the 3-ways (at each entrance to the LR) and all the 4-ways. The other two 3-ways control the living room lights. (I thought you said this was commercial, not that it matters.)
I agree also that you don't need the J boxes. I would feed the switches on the right of the drawing from the panel. From there run two sets of travellers from each 3-way + neutral to the 3 way on the left and all the 4-ways ending up at the 3-ways on the left. From those switches feed your lights which are simply tied together starting at one end of each group.
-Hal
