Contraction conjecture for the Lipschitz constants

For each $n$, let $y_n$ be the sequence-generated root and define

$$c_n(\varepsilon)=\frac{Z\left(\max_{t\in[0,y_n]}\{Y_{n+1,1}(t)\geqslant t\}+\epsilon\right)-Z\left(\min_{t\in[y_n,\infty]}\{Y_{n+1,1}(t)\leqslant t\}-\epsilon\right)}{2\varepsilon+\max_{t\in[0,y_n]}\{Y_{n+1,1}(t)\geqslant t\}-\min_{t\in[y_n,\infty]}\{Y_{n+1,1}(t)\leqslant t\}}.$$

This is the Lipschitz constant in the referenced formula. Contraction conjecture. It is always possible to choose a sufficiently small positive $\varepsilon$ such that

$$0<c_n(\varepsilon)<1.$$

Such a bound would provide a contraction estimate for the relevant Newton-iteration interval, but the supplied text gives no proof or resolution.