Constancy conjecture for cyclic solutions of the functional equations

Let $s\geq1$, let $0\leq\gamma<s$, and let ${\mathcal A}_{\gamma}$ be the analytic-function algebra considered in the source. Let $\mathcal S$ be the multiplicative semigroup generated by the integers $N_1,\ldots,N_s$, and let $g\in{\mathcal A}_{\gamma}$ be cyclic. Assume that, for every $N\in\mathcal S$, $g$ satisfies

$$g((z^N)^N)=\prod_{\zeta^N=1}g(\zeta z).$$

Constancy conjecture. Then $g$ is constant. The source presents this as a proposed extension of the assertion that non-constant outer solutions do not occur in the corresponding class; it does not provide a proof or a resolution.