The isomorphism conjecture for surface quiver groups and extended affine Weyl groups

Given an unpunctured surface $S$, let $Q$ be a quiver obtained from a triangulation of $S$. Let $G=G(Q)$ be the group generated by involutions $s_i$ with the defining relations associated to $Q$, and let $W=W({\bf u},Q)$ be the extended affine Weyl group generated by the reflections $r_i=r_{u_i}$ associated to a companion basis ${\bf u}$. The assignment $s_i\mapsto r_i$ induces a surjective homomorphism

$$\varphi:G(Q)\to W({\bf u},Q).$$

Isomorphism conjecture. The map $\varphi$ is an isomorphism, equivalently

$$G(Q)\cong W({\bf u},Q).$$

In finite and affine types the two groups are already known to be isomorphic; the conjecture asks whether this remains true for the groups arising from unpunctured surfaces and their associated extended affine Weyl groups.