Equality of characteristic polynomials for tropical and ordinary presentation matrices

Let $w\text{?}$ be a point of $b2X_{(t_0)}(b2R^{\mathrm{trop}})$, and let $b3$ be a path representing a mutation loop. For the sign sequence $b5_{b3}(w)$, consider the matrices $E^{b5_{b3}(w)}_{b3}$ and $\check{E}^{b5_{b3}(w)}_{b3}$. Characteristic-polynomial conjecture. Their characteristic polynomials agree up to an overall sign, and consequently their spectral radii are equal. The claim is supported by computer checks on many examples, with no counterexamples known; its formulation was suggested by Yuma Mizuno.