Uniform boundedness conjecture for the density of states at regular energies

Let $\rho$ be a probability density with moments of all orders, meaning that

$$\int |\omega|^q\rho(\omega)\,\mathrm d\omega<\infty$$

for every $q\geq 1$. Let $E_o\in\mathbb R$. Suppose there is a $\delta>0$ such that, on

$$\{\mathbf q:|\varepsilon(\mathbf q)-E_o|<\delta\},$$

the symbol $\varepsilon$ is $C^1$ and satisfies $\nabla\varepsilon(\mathbf q)\neq 0$. Uniform boundedness conjecture. There is a constant $C_\delta<\infty$ such that

$$\frac{\mathrm dN_\lambda(E)}{\mathrm dE}\leq C_\delta$$

for every $\lambda\in\mathbb R$ and every $E\in[E_o-\tfrac12\delta,E_o+\tfrac12\delta]$. This would strengthen the available regularity results by asserting a disorder-uniform bound on the density of states near any energy where the unperturbed dispersion relation has no critical points; the supplied text gives no resolution, so the conjecture is recorded as open.