AP abnormality for Cantor series normal numbers

Let $Q=(q_n)_{n\geq 1}$ be a basic sequence that is infinite in limit and fully divergent of type I. A real number is $Q$-normal if it belongs to $\mathscr{N}(Q)$, and is AP $Q$-abnormal if it does not belong to any of the sets $\mathscr{N}^{I}_{k,k,r}(Q)$ for $k\geq 2$ and $0\leq r\leq k-1$. AP abnormality conjecture. There exists such a sequence $Q$ and a real number $x$ such that

$$x\in\mathscr{N}(Q)\backslash\bigcup_{k=2}^{\infty}\bigcup_{r=0}^{k-1}\mathscr{N}^{I}_{k,k,r}(Q).$$

This predicts that ordinary $Q$-normality need not imply normality along every arithmetic progression. The paper presents it as a likely example of the distinction between these notions, but gives no proof.