Does regular action preserve cyclic subgroup generation?

Question

If you have three elements $x, y, z \in G$, it isn't hard to show that if $\langle y,z \rangle$ is cyclic, then $\langle x^{-1}yx, x^{-1}zx \rangle$ is cyclic. However, is it also true that $\langle yx, zx \rangle$ is cyclic?

Answer 1

Take $G = S_3$, and take $x = (12), \ y = e, \ z = (13).$ Then $\langle y,z \rangle = \langle (13) \rangle$, which is obviously cyclic.

However, $\langle yx,zx \rangle = \langle (12), (123) \rangle = S_3$, which is not cyclic.