The determinant and inverse conjecture for E0_2-matrices

Let $A\in\mathbb{R}^{n\times n}$ be an $\mathbf{E_{0_2}}$-matrix.

Determinant and inverse conjecture. One has

$$\det(A)<0,$$

$A^{-1}$ exists, and $A^{-1}$ is a $\mathbf{Z}$-matrix.

This is one of the conjectures proposed after examples and observations about $\mathbf{E_{0_2}}$-matrices. The supplied text gives no evidence of a proof or refutation, so its resolution remains open.