CAT(0) property for Helly groups

Determine whether Helly groups are CAT(0) groups.

Background

Helly groups admit proper cocompact actions on Helly graphs and hence on injective spaces. Whether every such group also admits a proper cocompact CAT(0) action is explicitly left open.

References

Are Helly groups CAT(0)?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 9