Positivity conjecture for fused multi-state ASEP update matrices

Let $M_{i,i+1}^{(\ell)}$ be the update matrix

$$M_{i,i+1}^{(\ell)}=\sum_{r=1}^{\ell}b_r^{(\ell)}M_{i,i+1}^{(\ell;r)}.$$

Normalize $b_r^{(\ell)}=\begin{bmatrix}\ell\\r\end{bmatrix}\widetilde b_r^{(\ell)}$, where

$$\begin{bmatrix}x\\y\end{bmatrix}=\frac{[x]!}{[x-y]![y]!},\qquad [x]!=[x][x-1]\cdots[1],\qquad [x]=\frac{q^x-q^{-x}}{q-q^{-1}}.$$

Because the update matrix is symmetric under $q\leftrightarrow q^{-1}$, take $0\leq q\leq1$. Positivity conjecture. The positivity conditions obtained from the matrix elements $\langle0|\otimes\langle r+x|M_{i,i+1}^{(\ell)}|r\rangle\otimes|x\rangle$, with $r=1,\dots,\ell-x$ and $x=0,\dots,\ell-1$, are

$$(-1)^{r-1}\widetilde b_r^{(\ell)}>0,$$

and

$$\sum_{k=0}^{s}q^{(s-1)k}\begin{bmatrix}s\\k\end{bmatrix}\widetilde b_{k+i}^{(\ell)}(-1)^{i-1}>0, \qquad i=1,\dots,\ell-s,$$

for $s=1,\dots,\ell-1$. This conjecture gives the parameter restrictions ensuring positivity of the arbitrary linear combination of fused Temperley–Lieb generators and hence the existence of the corresponding multi-state ASEP for arbitrary $\ell$.