Quadratic convergence of the refined Azuma-Hoeffding bound to the Bhattacharyya parameter

Let $\{U_k,\mathcal{F}_k\}_{k=0}^h$ be the martingale sequence introduced in the subsection, and let $Z_2^{(m)}$ denote the corresponding refined exponential base and $Z_{\text{B}}$ the Bhattacharyya parameter. Quadratic-convergence conjecture.

$$\lim_{m \rightarrow \infty} Z_2^{(m)} = Z_{\text{B}}$$

and this convergence is quadratic.

This conjecture is motivated by the numerical behavior observed for several discrete memoryless channels, where the sequence appears to converge rapidly to the Bhattacharyya parameter. The supplied source gives no evidence of a resolution.