Infinite-order separation conjecture for Cantor series

Suppose that $\mathbb{N}=A\mathbin{\dot\cup}B$ is a partition of the natural numbers. Let $Q$ be a basic sequence, and say that it is fully divergent when the relevant divergence condition holds for every order. Infinite-order separation conjecture. There exist a real number $x$ and a fully divergent basic sequence $Q$ such that

$$x\in\bigcap_{k\in A}\mathscr{N}_k(Q)\backslash\bigcap_{j\in B}\mathscr{N}_j(Q).$$

This would extend the finite-order separation assertion to an arbitrary partition of the natural numbers. The authors call it much more surprising and explicitly hesitate to suggest that it is true; it remains open.