Regularity of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes

Let $\Delta$ be a one-dimensional simplicial complex. Let $I_\Delta$ be the Stanley-Reisner ideal of $\Delta$. We prove that for all $s \ge 1$ and all intermediate ideals $J$ generated by $I_\Delta^s$ and some minimal generators of $I_\Delta^{(s)}$, we have $${\rm reg} J = {\rm reg} I_\Delta^s = {\rm reg} I_\Delta^{(s)}.$$