Stability conjecture for two-dimensional hyperbolic systems with directional viscosity

Let $A$, $B$, $C$, and $D$ be the matrices in the two-dimensional hyperbolic system (1.2), and define the stability operators

$$\Omega_1(k)=-ikA-k^2C,\qquad \Omega_2(k)=-ikA-k^2D,\qquad \Omega_3(k)=-ikB-k^2C,\qquad \Omega_4(k)=-ikB-k^2D.$$

Stability conjecture. If the eigenvalues of all four stability operators are negative for every wavenumber $k$, then the system (1.2) is stable.

The conjecture is motivated by combinatorial observations for one-dimensional systems and by unstable oblique waves. The supplied text gives no evidence that it has been resolved.