Markov Mallows processes with Markovian jumping chains

Let $n\geq 4$. A smooth Mallows process is a process

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

, with jumping process

$$(\widetilde{{\mathcal{M}}}_k)_{0\leq k\leq\binom{n}{2}}.$$

Let $\mathcal{T}^a$ and $\mathcal{T}$ denote the graph groups appearing in the source. Markovian-jumping-process conjecture. For every $n\geq 4$, there exist smooth Mallows processes ${\mathcal{M}}$ such that $({\mathcal{M}}_t)_{t\in[0,\infty)}$ is a Markov process and $(\widetilde{{\mathcal{M}}}_k)_{0\leq k\leq\binom{n}{2}}$ is a Markov chain. Every such process satisfies

$$\mathcal{T}^a\subsetneq\langle\mathcal{G}_{\mathcal{M}}\rangle\subseteq\mathcal{T},$$

and there exist such processes with

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

The source reports computational evidence for smooth, non-regular examples but gives no proof.