Four-head conjecture for finitely generated infinite groups

Let $G$ be a finitely generated infinite group. Write $h(G)$ for the supremum of the integers $n$ such that the $n$-head hierarchy has a strict inclusion at level $n$, and write $G \geq \mathbb{Z}$ for the condition that $G$ contains a subgroup isomorphic to $\mathbb{Z}$. The group $G$ has decidable word problem when there is an algorithm deciding whether a given word in its generators represents the identity.

Four-head conjecture. If $G \geq \mathbb{Z}$ and $G$ has decidable word problem, then $h(G)=3$; if $G \geq \mathbb{Z}$ and $G$ has undecidable word problem, then $h(G)=4$; and if $G$ is a torsion group, then $h(G)=\infty$.

This conjecture proposes that, among finitely generated infinite groups, the need for four heads is characterized by undecidability of the word problem for groups containing an infinite cyclic subgroup, while torsion groups have infinite head height. The paper establishes that $4$ occurs as a head height and that the only possible finite heights are $2$, $3$, and $4$, but the stated classification is not resolved here.