Helly property for uniform lattices in $GL(n,R)$

Determine whether uniform lattices in $GL(n,R)$ are Helly groups.

Background

Uniform lattices in GL(n,R)GL(n,R) are shown to act properly and cocompactly on injective metric spaces. The stronger conclusion that they admit such an action on a Helly graph remains unresolved.

References

Are uniform lattices in $GL(n,R)$ Helly? Note that biautomaticity is open for such groups.

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 6