In this talk, I will describe the current progress of the project and the main ideas behind the construction. The first step has been achieved by replacing each summand of a multisummable function with an almost regular germ, which yields the desired quasianalytic algebra G(Q).
I will also report progress on the second step. It turns out that the set of all convergent power series with coefficients in is too large to remain closed under certain substitutions. To address this issue, I restrict attention to those convergent power series whose asymptotic expansions satisfy a summability condition. I will explain how this subclass behaves well under a specific class of substitutions and is therefore suitable for the construction.