Conjecture on eigenvector independence for quasi-inverse matrices

Let $A$ be a nonsingular matrix, and let $A^\nabla$ denote its quasi-inverse matrix. Suppose that $A^\nabla$ has $n$ distinct eigenvalues. Quasi-inverse eigenvector conjecture. The eigenvectors corresponding to these $n$ distinct eigenvalues are independent. This is the complementary quasi-inverse statement introduced alongside the preceding conjecture; the supplied text gives no resolution status.