Existence of smooth Mallows processes with non-total graph structure

Let $n\geq 3$. A smooth Mallows process is a process

$${\mathcal{M}}={({\mathcal{M}}_t)_{t\in[0,\infty)}}$$

for which the associated graph group $\langle{\mathcal{G}}_{\mathcal{M}}\rangle$ is defined, and let $\mathcal{T}$ denote the full graph group. Existence conjecture. For every $n\geq 3$, there exists a smooth Mallows process ${\mathcal{M}}$ such that

$$\langle{\mathcal{G}}_{\mathcal{M}}\rangle\subsetneq{\mathcal{T}}.$$

This asks whether smooth Mallows processes can have strictly smaller graph structure than the full graph group; the source presents it as a possible direction for future work and gives no resolution.