Part of my hypothesis is that if you feed a TM location information, then it should remember sequences of locations. Said another way: L6 should represent the current location in the context of the previous locations. It makes sense to me that L6 uses efferent motor commands to update both the location and the context under which it got there. L6 is where path integration happens in my hypothesis.
Layer 5 recognizes sets of L6 cells which commonly activate near-in-time to each other, and assigns a static/stable set of L5 cells to represent them. L5 cells then respond to movement via L6 grids.