Farb's quasi-isometric rigidity conjecture for mapping class groups

Let $S$ be a non-exceptional surface of finite type, and let $\mathcal {MCG}(S)$ denote its mapping class group. Let $G$ be a finitely generated group that is quasi-isometric to $\mathcal {MCG}(S)$, meaning that their word-metric spaces are quasi-isometric. A homomorphism has finite kernel when its kernel is a finite group, and has finite index image when its image has finite index in the target group.

Farb's quasi-isometric rigidity conjecture. There exists a homomorphism

$$G\to\mathcal {MCG}(S)$$

with finite kernel and finite index image.

This is described in the source as a major outstanding conjecture concerning quasi-isometric rigidity of mapping class groups. It predicts that every finitely generated group quasi-isometric to a mapping class group is, up to finite kernel and finite index, algebraically represented by that mapping class group.