Normality equivalence conjecture for deterministic dynamically generated basic sequences

Let $Q=(q_n)_{n\geq 1}$ be a deterministic dynamically generated basic sequence. Write $\mathcal{N}(Q)$ for the set of $Q$-normal numbers, $\mathcal{UN}(Q)$ for the uniformly $Q$-normal numbers, $\mathcal{UDN}(Q)$ for the uniformly distribution-normal numbers, and $\mathcal{DN}(Q)$ for the distribution-normal numbers. Normality equivalence conjecture.

$$\mathcal{N}(Q)=\mathcal{UN}(Q)=\mathcal{UDN}(Q)=\mathcal{DN}(Q).$$

This conjecture would unify the principal normality notions for deterministic dynamically generated basic sequences; the preceding text says that the hotspot conjecture would imply it, while the full equivalence remains open.