The converse characterization of bases for asymptotically automatic sequences

Let $V$ be a vector subspace of $\bigoplus_{i=1}^\infty \mathbb{Q}$. For a prime $p$, let $\nu_p$ denote the $p$-adic valuation, and let $p_1,p_2,\dots$ be the increasing enumeration of the primes. Converse basis conjecture. There exists a sequence $\mathbf a$ over the alphabet $\{0,1\}$ such that, for every $k\in\mathbb{N}$ with $k>1$, $\mathbf a$ is asymptotically $k$-automatic if and only if

$$(\nu_{p_1}(k),\nu_{p_2}(k),\dots)\in V.$$

Corollary 1.5 gives the forward structural implication for every sequence; this conjecture asks whether every vector subspace can occur as the set of bases of a binary sequence. Its status is not resolved in the supplied source.