The layout from the video:

oox ooo ooo

Which means that you need 9 grids for every one slot. Further, and this is where a proof would be needed, you'd have the following robot placements:

roxrox rRRrRR rrrrRR roxrox rRRrRR rrrrRR

One robot can move in this case but I'm not sure this is optimal. At this density I'm betting that you get into exponential moves to move all the way from 0,0 to 6,6.

If you were more like roxrox orooro rorror roxrox orooro rorror

you could more likely keep channels of movement.

This is similar to a chip routing problem. So that's why you go vertical planes as well. But you still get limited throughput.

I may have the math wrong (I haven't done it)

EDIT: Ugh, no pre tags, take the ror ooo and line them up vertially on the breaks.