Mizuno's exponent characteristic-polynomial conjecture for Y-systems

Let $X_r$ be a finite type Dynkin diagram and let $\ell$ be a positive integer with $\ell greater than or equal to 2$. Let $J(\eta)$ be the Jacobian matrix at the unique $\eta in(\mathbb{R}_{>0})^{\mathbf{I}}$ satisfying $\mu(\eta)=\eta$, and let $N_{X_r,\ell}(x)$ and $D_{X_r,\ell}(x)$ be the polynomials defined from the root system of type $X_r$. Mizuno's exponent conjecture. The characteristic polynomial of $J(\eta)$ satisfies

$$\det(xI-J(\eta))=\frac{N_{X_r,\ell}(x)}{D_{X_r,\ell}(x)}.$$

This formula expresses the exponents of the Y-system through the root system. The source does not provide evidence of resolution for the general conjecture; it is proved only in the special cases $(A_1,\ell)$ and $(A_r,2)$.