One-apart independence conjecture for multispecies TASEP correlations

Let $w=(w_a)$ denote the cyclic multispecies TASEP word, and let $E_{i_1,\dots,i_r}$ denote the joint probability of seeing $i_a$ in position $a$, for $a=1,\dots,r$. One-apart independence conjecture. For any $i_1,i_2,\dots,i_r$ and $b,k$ satisfying

$$k>b-r+\max_a i_a,$$

we have

$$\operatorname{P}(w_a=i_a\text{ for }1\leq a\leq r,\text{ and }w_b=k)=\frac{1}{n}E_{i_1,\dots,i_r}.$$

This generalizes the preceding factorization to a particle in a more distant position and is motivated by uniformity results for particles farther apart. The source offers it as an open conjecture without a proof.