Constancy conjecture for the integrated density of states at inverse-beta energies

Let $p$ be the Bernoulli parameter, let $\beta^{-1}(n)$ denote the decreasing sequence appearing in the model, and let $I_{p,\zeta}(x)$ be the integrated density of states at disorder parameter $\zeta$. For $n\in\mathbb{N}\setminus\{1\}$, consider the energy $x=\beta^{-1}(n)$. Constancy conjecture. For every $n\in\mathbb{N}\setminus\{1\}$ and every $\zeta\geq\beta^{-1}(n)$,

$$I_{p,\zeta}(\beta^{-1}(n))=\frac{p}{1-p}\frac{(1-p)^n}{1-(1-p)^n}.$$

This conjecture asserts that the integrated density of states is constant in $\zeta$ on the indicated range, extending the analogous constancy established in the preceding corollary for the energies $4-\beta^{-1}(n)$. The supplied text gives no resolution status.