Existence of exceptional polynomial bases with equidistributed random zeros

Let $E\subset\mathbb{C}$ have non-analytic boundary, and let $\{A_k\}_{k=0}^\infty$ be i.i.d. random variables with $\mathbb{E}[\log^+|A_0|]=\infty$. For a basis $\{B_k\}_{k=0}^\infty$ of Szegő, Bergman, or Faber polynomials, define

$$P_n(z)=\sum_{k=0}^n A_kB_k(z).$$

Exceptional-basis conjecture. It is possible to construct such sets $E$ and sequences of random variables such that, for each of these polynomial bases, some subsequence of zero counting measures satisfies

$$\tau_{n_k}\stackrel{w}{\longrightarrow}\mu_E\quad\text{almost surely}.$$

This conjecture concerns the apparent failure of the necessity direction for almost-sure zero equidistribution under infinite logarithmic moment. The surrounding discussion indicates that the general necessity question is open, while Proposition 2.3 provides a related construction for the monomial basis.