I think you misunderstood the paper. Grid cells don't encode location, a set of grid cell modules do. So there isn't a grandmother cell here at all. He's simply proposing this theory because of the noted presence of grid cells in the neocortex.

His new theory, if I understand it correctly, is pretty basic. It is simply saying every sparse encoding of sensory inputs is also tied to a spatial encoding by sets of grid cell modules within the same cortical column. For deep learning practitioners, it would be like saying every embedding comes with it's own spatial encoding, or placed into the same embedding space, however you want to look at it.

These spatial encodings are then related to other spatial encodings of other sensory inputs on different cortical column stacks (by displacement cells?), to firstly build up representations of objects, and then, he surmises further, the relationships of these objects with each other (part to whole).

I think the last part is probably what Jeff will need to substantiate a bit more.