Multiline-queue Markov-chain conjecture for inhomogeneous multispecies TASEP weights

Let $\mathcal Q$ be a multiline queue with state space $\Omega^{FM}_m$, and let $\Omega_m$ be the state space of the corresponding inhomogeneous multispecies exclusion process. For $1\leq i<r\leq n$, let $z_{r,i}(\mathcal Q)$ be the number of vacancies on row $r$ that are $i$-covered by bully paths. Let $v_r=N-M_r$ be the number of vacancies on row $r$, and define

$$V_r=\sum_{i=r+1}^{n-1}v_i.$$

Let $B$ denote the bully-path projection from multiline queues to particle configurations. Multiline-queue Markov-chain conjecture. There is a Markov chain on $\Omega^{FM}_m$ which lumps via $B$ to the inhomogeneous multispecies exclusion process on $\Omega_m$, such that the stationary weight of every configuration $\mathcal Q$ is

$$w(\mathcal Q)=x_1^{V_1}x_2^{V_2}\dots x_{n-2}^{V_{n-2}}\prod_{1\leq i<r\leq n}\left(\frac{x_r}{x_i}\right)^{z_{r,i}}.$$

This conjecture would give a combinatorial realization of the Lam–Williams stationary weights and extend them to multipermutations. The paper reports partial results supporting it, but says that it has not been settled in full generality.