Generic asymptotic eigenvalue-counting conjecture for piecewise continuous elliptic systems

Let $L f(x)=A(x)f'(x)$ on $L^2((\alpha,\beta),\mathbf C^n)$, where $A(x)$ is invertible and diagonalizable for each $x\in(\alpha,\beta)$, with boundary conditions $f(\alpha)=u$ and $\langle f(\beta),v\rangle=0$ for nonzero $u,v\in\mathbf C^n$. For each $x$, let $K(x)$ be the convex hull of the eigenvalues of $A(x)^{-1}$, and let $b(x)$ be the length of its boundary. Generic asymptotic eigenvalue-counting conjecture. The asymptotic formula

$$N(E)=\frac{E}{2\pi}\int_\alpha^\beta b(x)\,{\rm d}x+O(1)$$

as $E\to\infty$, which holds generically for piecewise constant coefficients, remains generically valid for piecewise continuous coefficients $A(x)$ on $[\alpha,\beta]$. The claim extends the piecewise-constant result to variable coefficients; the supplied text gives no resolution, so its status remains open.