The rank conjecture for groups recognised by Cho automata

Let $H$ be a group and let $k$ be a nonnegative integer. A $k$-counter Cho automaton is a Cho automaton with $k$ counters whose accepted language includes the word problem of $H$. A group is virtually free abelian of rank $k$ if it has a finite-index free abelian subgroup of rank $k$.

Rank conjecture. If the word problem of $H$ is accepted by a $2k$-counter or $(2k+1)$-counter Cho automaton, then $H$ is virtually free abelian of rank $k$.

This is the proposed strengthening of the lower-bound statement discussed immediately beforehand: the known argument gives virtual free abelianness of rank at most the counter parameter, while this conjecture predicts an exact rank under the stated $2k$- or $(2k+1)$-counter hypothesis. The source gives no resolution.