Algebraic criterion for stability of a viscous hyperbolic operator

Let $A$ be a matrix with distinct real eigenvalues $\lambda_\ell$, for $\ell=1,\ldots,N$, and let $C$ be a matrix with real positive eigenvalues. Define

$$\Omega(k)=-ikA-k^2C.$$

For each $\ell$, let $\lambda_{\ell,1}$ denote the quantity defined by the first-order correction formula (5.2).

Algebraic stability conjecture. The operator $\Omega(k)$ has negative eigenvalues if and only if $\lambda_{\ell,1}<0$ for every $\ell=1,\ldots,N$.

This conjecture is presented as an algebraic reduction of the preceding stability conjecture. The supplied text gives no evidence that it has been resolved.