TASEP-limit conjecture for the multi-state ASEP

Let $M_{i,i+1}^{(\ell)}$ be the multi-state ASEP update matrix formed from the fused generators $M_{i,i+1}^{(\ell;r)}$. TASEP-limit conjecture. The reparametrization

M_{i,i+1}^{(\ell)}=\sum_{r=1}^{\ell}q^{r\ell}\begin{bmatrix}\ell\r\end{bmatrix}\widetilde b_r^{(\ell)}M_{i,i+1}^{(\ell;r)}

gives a proper TASEP limit. The resulting hopping rates are

$$w(i,j\mid i-x,j+x)=(-1)^x\widetilde b_x^{(\ell)},\qquad x=1,\dots,\ell,$$

and positivity requires $0<\widetilde b_x^{(\ell)}<1$ for even $x$ and $-1<\widetilde b_x^{(\ell)}<0$ for odd $x$. This conjecture concerns the well-definedness of the $(\ell+1)$-state TASEP obtained in the $q\to0$ limit of the multi-state ASEP.