Individual Volterra–Lyapunov stability implies a non-square D-stability-like property

Let $A\in\mathbb{R}^{m\times n}$ be a real nonsquare matrix. Let $\mathcal{M}$ be the index set consisting of $k$-tuples of integers in the range $1,\ldots,m$. For $j\in\mathcal{M}$, let $[AEK]_j$ denote the corresponding matrix associated with the block diagonal non-square matrix $K\in\mathbb{C}^{n\times m}$, and let $\operatorname{Re}\{\sigma_i(M)\}$ denote the real part of the $i$-th eigenvalue of a matrix $M$. Individual Volterra–Lyapunov stability conjecture. If $A$ is an individual Volterra–Lyapunov stable matrix, then there exists a block diagonal non-square matrix $K\in\mathbb{C}^{n\times m}$ such that, for every non-negative diagonal matrix $E\in\mathbb{R}^{n\times n}$ and every $j\in\mathcal{M}$,

$$\operatorname{Re}\{\sigma_i([AEK]_j)\}>0.$$

The claim proposes a sufficient spectral property associated with individual Volterra–Lyapunov stability for nonsquare matrices; the supplied text does not establish it or provide evidence that it has been resolved.