The relation-number formulation of the Main Conjecture

A rational number $q$ is called an $\ell$-step relation number if it satisfies the corresponding finite-step relation condition from the paper, and a relation number if some nontrivial word in the generators becomes the identity after substituting $a$ and $b_q$. Here $\ell\geq 0$.

Relation-number conjecture. Every rational number in $(-4,4)$ is an $\ell$-step relation number for some $\ell\geq 0$.

By the Lyndon–Ullman lemma, this is a diophantine-type reformulation of the Main Conjecture: being an $\ell$-step relation number for some $\ell$ is equivalent to being a relation number, hence to non-freeness of the generated group. The general assertion is not proved in the paper.