I used to do electroplating research.

This three-year-old article is pretty useful for informing yourself about aluminum-ion batteries.

This project isn't even listed on the head guy's research group site: http://dailab.stanford.edu/research.htm

Furthermore, he has no previous papers involving aluminum batteries. As such, this falls out of his area of expertise, which appears to primarily be graphene and carbon nanotubes. No doubt he's a respected scientist, but everything looks like roses when you're first starting down a new research path. In fact, given his previous expertise, when they say "graphite cathodes" and "a few types of graphite", given previous research by Cornell using vanadium oxide nanotubes, I suspect the Stanford team is using carbon nanotubes in their cathode. Who says a few types of graphite? Nobody. They are referring to graphene. At which point you're all free to laugh. Using graphite isn't particularly novel in any case, it's a known solution for aluminum-air batteries and lithium-ion batteries, but with regards to the aluminum-air batteries they are completely impractical because the cathodes are extremely complex and use expensive materials and have only found military use, where cost is no object.

As such it seems like the standard failure to research the failures of the people whose footsteps you're following in. Theoretical physical constraints are seductive, the feats of engineering required to approach them are the slayers of men. The Oak Ridge National Laboratory and Cornell have been working on aluminum batteries for years and Stanford has only just started.

I'm afraid that even if all the other claims are 100% true, this is yet another technology waiting for graphene and carbon nanotubes to be cheaply produced at industrial scales.

The second core issue (besides the necessity of ionic liquids) is that the aluminum anode will naturally corrode on normal operation, and all possible solutions have involved complex and expensive cathodes to mitigate the problem and this seems no different.