Injective hulls of hyperbolic spaces

Determine the injective hull of hyperbolic $n$-space.

Background

The notes explicitly ask for a description extending the unresolved two-dimensional hyperbolic-plane case to higher-dimensional hyperbolic spaces.

References

What is the injective hull of the hyperbolic $n$-space?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 5