  We solve the regularity problem for Milnor's infinite dimensional Lie groups
in the asymptotic estimate context. Specifically, let $G$ be a Lie group with
asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by
$\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow
G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that
$\mathrm{evol}$ is $C^\infty$-continuous on $\mathrm{D}\cap
C^\infty([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is
$C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore
show that $G$ is k-confined for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$
if $G$ is constricted. (The latter condition is slightly less restrictive than
to be asymptotic estimate.) Results obtained in a previous paper then imply
that an asymptotic estimate Lie group $G$ is $C^\infty$-regular if and only if
it is Mackey complete, locally $\mu$-convex, and has Mackey complete Lie
algebra - In this case, $G$ is $C^k$-regular for each $k\in \mathbb{N}_{\geq
1}\sqcup\{\mathrm{lip},\infty\}$ (with ``smoothness restrictions'' for
$k\equiv\mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially
complete with integral complete Lie algebra.
