The free-subgroup conjecture for the Manturov group $G^2_4$

Let $G^2_4$ be the group generated by $a,b,c,d,e,f$ with the defining relations given in the paper.

Free-subgroup conjecture. The group $G^2_4$ has a free subgroup of rank $2$.

The authors state this conjecture because the growth of $G^2_4$ is currently unknown; establishing a rank-two free subgroup would provide information about its structure and growth.