Conjecture on dependent eigenvectors and the quasi-inverse matrix

Let $A$ be a nonsingular matrix with $n$ distinct eigenvalues. Suppose that the eigenvectors of $A$ are dependent, and let $A^\nabla$ denote its quasi-inverse matrix. Write $f_{A^\nabla}(x)$ for the characteristic polynomial of $A^\nabla$ and $f^{^{es}}_{A^\nabla}$ for its essential characteristic polynomial. Quasi-inverse conjecture. Under these hypotheses, (1) $\det(A)f_{A^\nabla}(x)$ strictly ghost-surpasses $x^nf_A(x^{-1})$; (2) if $f_{A^\nabla}\ne f^{^{es}}_{A^\nabla}$, then $A^\nabla$ has fewer distinct eigenvalues than $A$; and (3) the eigenvectors of $A^\nabla$ are independent. The conjecture concerns how dependence among the eigenvectors of a nonsingular matrix is reflected by its quasi-inverse; the supplied text gives no resolution status.