Closure under composition?

The Ilyashenko field we construct in our most recent paper is not closed under composition, or even under $\log$-composition.
How do we know this? By construction, every LE-series that is the asymptotic expansion of a germ $f$ in the Ilyashenko field $\K$ is an LE-series with convergent LE-monomials, but in general the series is divergent. Since the infinite part of such a series has, in general, infinite support, the germ $\exp(f)$ has an LE-series as asymptotic expansion in at least one LE-monomial that is not convergent; hence $\exp(f)$ cannot belong to $\K$.
However, $\K$ is closed under composition on the right with infinitely increasing $\log$-$\exp$-analytic germs. We did not include the proof of this claim in our paper, as the latter is already long enough, but we will need this in our subsequent work of defining corresponding Ilyashenko algebras in several variables.

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