Injective hull of the hyperbolic plane

Determine the injective hull $E(H^2)$ of the hyperbolic plane $H^2=SL(2,R)/SO(2)$.

Background

The injective hull of the hyperbolic plane is known to lie at bounded distance from the subspace of volume-one ellipses inside the injective space of convex bodies, but its precise structure is not determined.

References

What is the injective hull $E(H2)$ of the hyperbolic plane $H2=SL(2,R)/SO(2)$?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 4