Mizuno's exponent formula for finite type Y-systems

Let $X_n$ be a finite type Dynkin diagram and let $\ell$ be a positive integer with $\ell\geq 2$. Let $\gamma=\gamma(X_n,\ell)$ be the mutation loop on the quiver $Q(X_n,\ell)$. The exponents are the integers determined by the eigenvalues of $J_\gamma(\eta)$, and the polynomials $N_{X_n,\ell}(z)$ and $D_{X_n,\ell}(z)$ are defined by

$$N_{X_n,\ell}(z)=\prod_{i=1}^n\frac{z^{t(\ell+h^\vee)}-1}{z^{t/t_i}-1},\qquad D_{X_n,\ell}(z)=D_{X_n,\ell}^{\rm long}(z)D_{X_n,\ell}^{\rm short}(z).$$

Mizuno's conjecture. The following identity holds:

$$\det (zI-J_\gamma(\eta))=\frac{N_{X_n,\ell}(z)}{D_{X_n,\ell}(z)}.$$

This conjectural formula describes the characteristic polynomial of the Jacobian at the positive fixed point in terms of the root data of the finite type Dynkin diagram. The source attributes the conjecture to Mizuno; no resolution is supplied here.