Nilpotent groups and 2-generated subgroups

Do you know an example of a $2$-locally nipotent group $G$ which is not locally nilpotent?

$2$-locally nilpotent: every subgroup which is generated by $2$ elements is nilpotent.

locally nilpotent: every finitely generated subgroup is nilpotent.

The groups referred to in YCor's answer to this question are infinite $d$-generator $p$-groups in which every $(d-1)$-generator subgroup is finite, and hence nilpotent since it is a $p$-group. So they provide examples, although it might be hard to find out much about them.
• More generally a finitely generated solvable torsion group is finite. You can prove it by induction on the derived length. $G/G'$ is finitely generated abelian and hence finite, so $G'$ is finitely generated, and the result follows by induction. – Derek Holt Jun 22 '18 at 19:40
• @user1729 It's an exercise to show that every infinite finitely generated solvable group has a finite index subgroup having $\mathbf{Z}$ as quotient. – YCor Jun 23 '18 at 10:08