Far-particle factorization conjecture for multispecies TASEP correlations

Let $E_{i_1,\dots,i_r}$ denote the joint probability of seeing particle types $i_a$ in positions $a$, for $a=1,\dots,r$. Far-particle factorization conjecture. For any $i_1,i_2,\dots,i_r$ and any $k$ satisfying

$$k>1+\max_a i_a,$$

we have

$$E_{i_1,\dots,i_r,k}=\frac{1}{n}E_{i_1,\dots,i_r}.$$

This is presented as a generalization of the three-point relation in which a sufficiently separated final particle contributes an independent factor $1/n$. The source gives no proof and relates it to a finite cyclic version of a result cited there as Lemma 7.3.