Two-block independence conjecture for multispecies TASEP correlations

Let $w=(w_a)$ denote the cyclic multispecies TASEP word, and let $E_{i_1,\dots,i_r}$ and $E_{j_{r+1},\dots,j_{r+s}}$ denote the corresponding nearest-neighbour block probabilities. Two-block independence conjecture. For any $i_1,\dots,i_r,j_{r+1},\dots,j_{r+s}$ such that, for every $b$ with $r+1\leq b\leq r+s$,

$$\min_b j_b>1+\max_a i_a,$$

we have

$$\operatorname{P}(w_a=i_a,\ 1\leq a\leq r,\text{ and }w_b=j_b,\ r+1\leq b\leq r+s)=E_{i_1,\dots,i_r}E_{j_{r+1},\dots,j_{r+s}}.$$

This is the source's most general proposed independence statement for two separated blocks of nearest-neighbour correlations. It is stated as an open conjecture, with no proof supplied.