Convergence of cascade tree decomposition KL divergence to zero

Let $\underline{X}$ be the input random vector and let $\underline{X}_{{\mathcal M}_i}$ denote the model random vector obtained after $i$ cascade-tree stages. Write ${\mathcal D}(f_{\underline{X}}(\underline{x})\,||\,f_{\underline{X}_{{\mathcal M}_i}}(\underline{x}))$ for the Kullback–Leibler divergence between their distributions. Convergence conjecture. The KL divergence converges to $0$ as the number of cascade trees $i$ increases. The preceding theorem establishes only that this divergence decreases and converges to a finite value; whether the limiting value is $0$ is the conjectured strengthening.