Characterization of inverse M-matrix interval systems

Let $\mathbf A=[\underline{A},\overline{A}]\in\mathbb{IR}^{n\times n}$ be an interval matrix, with midpoint $\operatorname{Mid}(A)$ and radius $\operatorname{Rad}(A)$, and let $z^i=(1,\dots,1,-1,1,\dots,1)^T$ have $-1$ in the $i$th entry and $1$ elsewhere. An interval matrix is an inverse M-matrix when it is nonsingular and its inverse is an M-matrix.

Inverse M-matrix characterization. $\mathbf A$ is an inverse M-matrix if and only if

$$\operatorname{Mid}(A)\pm\operatorname{diag}(z^i)\operatorname{Rad}(A)\operatorname{diag}(z^j)$$

for $i,j=1,\dots,n$, are inverse M-matrices.

This would provide a characterization involving only $2n^2$ ordinary matrices, potentially yielding an efficient test for whether an interval matrix is an inverse M-matrix. The computational complexity of this recognition problem is stated in the surrounding discussion to be open; no polynomial reduction or NP-hardness result is known there.