Finite-tower bounds for Skolem functions

We bound the eventual order types of Skolem functions below finite exponential towers. Writing $E_0(u)=u$, $E_{n+1}(u)=2^{E_n(u)}$, and $ω_0=1$, $ω_{k+1}=ω^{ω_k}$, the argument gives \[ |\Sk_{<E_n(x^m)}|<ω_{r_n},\qquad r_n=2+\frac{n(n+3)}2\quad(n\ge1,\ m\ge2\text{ fixed}). \] For triple towers w