Singular-component conjecture for A-L measures on Dirichlet-regular Widom sets

Let ${\mathsf{E}}$ be a Dirichlet-regular Widom set with DCT, and let $\mu^\alpha$ be the associated measure for some $\alpha\in\pi_1(\Omega)^*$. The A-L condition means that the complex Martin function $\Theta$ is normalized so that

$$\lim_{y\to\infty}\frac{\operatorname{Im}\Theta(iy)}{y}=1.$$

Conjecture on singular A-L measures. There exists a Dirichlet-regular Widom set ${\mathsf{E}}$ with DCT such that A-L holds and $\mu^\alpha$ has a nontrivial singular component with respect to Lebesgue measure for some $\alpha$.

The preceding lemma shows that under A-L, Lebesgue measure is absolutely continuous with respect to $\mu^\alpha$, so the conjecture asks whether the converse can fail through a singular component of $\mu^\alpha$.