Singular-spectrum conjecture for Anderson models on antitrees

Let ${\mathbf s}$ be a sequence and let $H_\lambda={\mathcal A}_{\mathbf s}+\lambda {\mathcal V}$ be the Anderson model on the antitree ${\mathbb A}_{\mathbf s}$, with disorder parameter $\lambda$. Singular-spectrum conjecture. If

$$\sum_n s_n^{-1}=\infty,$$

then the spectrum of $H_\lambda$ is almost surely singular at any disorder. This conjecture predicts the absence of absolutely continuous spectrum when the reciprocal growth rates are not summable, for every disorder strength. Its status is not resolved in the supplied text.