Boundary conjecture for the analytic filtration

Let $\mathfrak A=\{\mathfrak A_\alpha\}_{\alpha\le1}$ be the analytic filtration, and let $F_\alpha$ denote the function used in its definition. For each $\alpha\le1$, consider the boundary $\partial\mathfrak A_\alpha$ and the rotations of $F_\alpha$ given by $zF_\alpha(e^{i\theta}z)$, where $\theta\in\mathbb R$. Boundary conjecture. The boundary $\partial\mathfrak A_\alpha$ coincides with

$$\left\{zF_\alpha(e^{i\theta}z):\theta\in\mathbb R\right\}$$

for every $\alpha\le1$. This describes the entire boundary of each class in the analytic filtration; the source presents it as an assumption, and no resolution is supplied.