Injective uniform lattices in semisimple Lie groups

Classify the uniform lattices in semisimple Lie groups that act properly and cocompactly on injective metric spaces.

Background

The notes establish injective actions for certain uniform lattices in classical groups and indicate that only partial answers are available in the cited work. A general classification is explicitly requested.

References

Which uniform lattices in semisimple Lie groups are injective? There are partial answers in.

Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 7