The only other case that is known is n=24: Abhinav Kumar and I solved that case (Optimality and uniqueness of the Leech lattice among lattices, Annals of Mathematics 170 (2009), 1003-1050, doi:10.4007/annals.2009.170.1003, arXiv:math.MG/0403263).
It might be possible to do n=9 using known methods, but it would be an enormous calculation. For the status as of a few years ago, see the end of Mathieu Dutour Sikiric, Achill Schuermann, and Frank Vallentin's paper
- Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21-32, doi:10.1090/S1079-6762-07-00171-0, arXiv:math/0609388.
