Characteristic-polynomial reversal conjecture for pseudo-inverses of supertropical matrices

Let $R$ be the supertropical semiring, let $A=(a_{i,j})\in M_n(R)$ be a non-singular matrix, and let $A^\nabla$ denote its pseudo-inverse. Write

$$f_A(x)=\sum_{k=0}^n a_kx^k,\qquad f_{A^\nabla}(x)=\sum_{k=0}^n b_kx^k,$$

where $\models_{gs}$ denotes ghost surpassion and $\det(A)$ is the determinant of $A$.

Characteristic-polynomial reversal conjecture. One has

$$\det(A)f_{A^\nabla}(x)\models_{gs} x^nf_A(x^{-1}).$$

Equivalently,

$$\det(A)b_k\models_{gs} a_{n-k}\qquad\forall k=0,\ldots,n.$$

This predicts that the characteristic polynomial of the pseudo-inverse has, after multiplication by $\det(A)$, coefficients related by ghost surpassion to the reversed coefficients of the characteristic polynomial of $A$, extending the behavior seen in the motivating example and relating the supertropical eigenvalues of $A$ and $A^\nabla$.