Increasing nearest-neighbour correlation conjecture for the multispecies TASEP

Let $E_{i_1,\dots,i_r}$ denote the joint probability of seeing particle types $i_a$ in positions $a$, for $a=1,\dots,r$. Increasing-correlation conjecture. If

$$i_1<i_2-1<i_3-2<\dots<i_r-(r-1),$$

then

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

This extends the observed $1/n^3$ formula for the simplest increasing three-point correlation. A proof was not available in the source, which describes the formula as resistant to the paper's methods.