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](https://doi.org/10.4007/annals.2009.170.1003), arXiv:[math.MG/0403263](https://arxiv.org/abs/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](https://doi.org/10.1090/S1079-6762-07-00171-0), arXiv:[math/0609388](https://arxiv.org/abs/math/0609388).