Generalized Lam–Williams positivity conjecture for multispecies TASEP components

Let $\boldsymbol\tau$ and $\boldsymbol\nu$ be the model parameters, let $\psi_{\bf w}(\boldsymbol\tau,\boldsymbol\nu)$ denote the component associated with a particle configuration $\bf w$, and normalize the vector of components by

$$\psi_{12\dots N}(\boldsymbol\tau,\boldsymbol\nu)=\phi_N(\boldsymbol\tau,\boldsymbol\nu),$$

where

$$\phi_N(\boldsymbol\tau,\boldsymbol\nu)=\prod_{1\leq \alpha<\beta\leq N}(\tau_\alpha-\nu_\beta)^{\beta-\alpha-1}.$$

Here $\boldsymbol\nu$ is replaced by $-\boldsymbol\nu$ in the components appearing in the claim. Generalized Lam–Williams positivity conjecture. With this normalization, the components $\psi_{\bf w}(\boldsymbol\tau,-\boldsymbol\nu)$ are polynomials in $\boldsymbol\tau$ and $\boldsymbol\nu$, with positive integer coefficients. The source presents this as a conjecture suggested by numerical computations at small sizes, extending the positivity phenomenon of Lam and Williams to generic $\boldsymbol\nu$; no resolution is supplied in the excerpt.