For the same energy, the mass of an aluminum electrode would need to be almost 2.4 times greater than the mass of a lithium electrode, in a sulfur-based battery (the ratio can be computed from the enthalpies of the 2 sulfides, the atomic masses of Li and Al, and their number of valence electrons).
However, while there are also lithium-sulfur batteries in development, the current lithium-ion rechargeable batteries with Co/Ni/Mn/Fe electrodes have a much worse energy/mass ratio than a lithium-sulfur battery.
Moreover, in a lithium-ion battery, the mass of the electrode which stores the lithium is much greater than the mass of the stored lithium (which is intercalated in a porous structure), and the mass of the electrode is only a small fraction of the total mass of the battery.
The proposed aluminum-sulfur battery needs good thermal insulation, which will increase the volume in comparison with a lithium battery, but which should not increase much the mass.
In conclusion, it is likely that it should be possible to make such a battery at a similar energy per mass with the current Li-ion batteries, but at a worse energy per volume.
There is a chance to improve the energy per volume by making a very large battery, which would be possible because there is less risk of fires, but in a large battery the regrowth of the aluminum might be not uniform enough, resulting in a shorter number of charge-discharge cycles until degradation.
