Polynomial closure conjecture for the sets $\mathcal{A}_n$

For each integer $n\geq 1$, let $\mathcal{A}_n$ be the set defined in the paper, viewed as a subset of the algebraic integers $\overline{\mathbb{Z}}$. A subset is polynomially closed in $\overline{\mathbb{Z}}$ if it equals its polynomial closure there. Polynomial closure conjecture. For each $n\geq 1$, $\mathcal{A}_n$ is polynomially closed in $\overline{\mathbb{Z}}$. The paper proves the case $n=1$, while the cases $n\geq 2$ remain open in the supplied context.