Subexponential Lebesgue constants for pseudo Leja sequences

Let $E$ be a regular compact set in $\mathbb{C}$, let $(a_n)_{n=0}^{\infty}$ be a pseudo Leja sequence of bounded Edrei growth in $\partial \widehat{E}$, and let $\Lambda_n(E,a^{(n)})$ denote the corresponding Lebesgue constant, where $a^{(n)}=\{a_0,\ldots,a_n\}$. Pseudo Leja Lebesgue-constant conjecture. The condition

$$\lim_{n\rightarrow\infty}\Lambda_n(E,a^{(n)})^{1/n}=1$$

holds for every such $E$ and every such pseudo Leja sequence. The result is known when the outer boundary is a finite union of quasiconformal arcs, and for sequences arising from Leja sequences on a circle via the Riemann map; the assertion for arbitrary regular compact sets and bounded Edrei growth remains open.