I have proposed a positive solution to this problem in a preprint entitled Hyperexponentially closed fields, to be found here, more precisely in Sections 10.1 and 10.2.
The solution is based on work of G. Edgar and J. Ecalle, where one uses formal properties of generalized power series (e.g. transseries) to keep track of how conjugating elements $\varphi$ in $\varphi f = g \varphi$ behave as functions $h \mapsto \varphi h$, very roughly speaking.
