Niv's characteristic-polynomial conjecture for supertropical adjoint matrices

Let $(\mathcal{G},\ast,0,\leq)$ be an ordered Abelian group, let $\mathcal{S}=\mathcal{G}^{(0)}\cup\mathcal{G}^{(1)}\cup\{\varepsilon\}$ be the associated supertropical semifield, and let $A\in\mathcal{S}^{n\times n}$ be non-singular. Write $\det_{\circ}A$ for the supertropical determinant, $\operatorname{adj}_{\circ}A$ for the adjoint matrix, and $\chi^k_{\circ}(A)$ for the supertropical sum of the principal $k\times k$ minors of $A$. For $c,d\in\mathcal{S}$, write $c\models d$ when either $c=d$ or $c=d+g$ for some ghost element $g$. Niv's conjecture. For every $k\in\{0,\ldots,n\}$,

$$\chi^k_{\circ}(\operatorname{adj}_{\circ}A)\models\left(\det_{\circ}A\right)^{\odot(k-1)}\odot\chi^{n-k}_{\circ}(A).$$

This conjecture proposes a characteristic-polynomial relation for the adjoint of a non-singular supertropical matrix, using ghost surpassing in place of equality; the supplied source identifies it as an open problem.