Uniform normality preservation under rational multiplication for dynamically generated basic sequences

Let $Q=(q_n)_{n\geq 1}$ be a dynamically generated basic sequence, and let $\mathcal{UN}(Q)$ denote the set of uniformly $Q$-normal numbers. Uniform normality multiplication conjecture. If $x\in\mathcal{UN}(Q)$ and $r\in\mathbb{Q}$, then $rx\in\mathcal{UN}(Q)$; equivalently, $\mathcal{UN}(Q)$ is preserved under multiplication by every rational number. The paper has established the corresponding rational-multiplication result in another normality setting, but uniform normality preservation remains conjectural here.