Lam–Williams positivity conjecture for inhomogeneous multispecies TASEP weights

Let the inhomogeneous multispecies exclusion process have $n$ classes of particles, with particles of class $i$ jumping at rate $x_i$. Let $w(\boldsymbol{\rho})$ denote the stationary weight of a configuration $\boldsymbol{\rho}$, and let $w_0=n,n-1,\dots,2,1$ be the reverse permutation. Lam–Williams positivity conjecture. With the normalization

$$w(w_0)=x_1^{\binom{n-1}{2}}x_2^{\binom{n-2}{2}}\dots x_{n-2}^{\binom{2}{2}},$$

every stationary weight $w(\pi)$ is a polynomial in the jump rates with positive integer coefficients. Lam and Williams formulated this conjecture for the inhomogeneous multispecies TASEP; the paper notes that it remains unresolved in full generality, although special cases are proved.