Subgroup of the braid group $B_n$ generated by $\sigma_i$ and $\sigma_{i+1} ^2$ is a free group?

Question

Let $B_n$ be the braid group on $n$ strands, $n \geq 3$. Consider a subgroup $H$ generated by $\sigma_i$ and $\sigma_{i+1}^2$. Is $H$ a free group of rank $2$?

Answer 1

No. Let $a=\sigma_i$ and $b=\sigma_{i+1}$. Then $b^2ab^2a^{-1}b^{-2}a^{-1} b^{-2}a=1$.

The group generated by the squares $\sigma_{i}^2$ and $\sigma_{i+1}^2$ is known to be free.