The upper-right two-point correlation conjecture for continuous TASEP

For a continuous TASEP on a ring with $n$ particles, let

$$c_{i,j}(n)=\operatorname{\mathbb P}(w_a=i,w_{a+1}=j)$$

be the stationary two-point correlation for an adjacent pair, where $a$ is arbitrary. The upper-right correlation conjecture. For every $i+1<j$,

$$c_{i,j}(n)=\frac{n}{\binom{n+j}{2}}.$$

The conjecture is motivated by computed correlation tables for $n\le 6$ and describes the apparently constant upper-right part of the correlation matrix. The paper also states a stronger conjecture covering all pairs, which subsumes this one.