Tits alternative for Helly groups

Determine whether Helly groups satisfy a Tits alternative.

Background

Helly groups have strong algorithmic and geometric properties, but the notes identify the Tits alternative as an unresolved structural question for this class.

References

Do Helly groups satisfy a Tits alternative?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 8