The accelero-summation conjecture for a composition-preserving Hardy field

Let $\mathbb{T}^{\operatorname{da}}$ denote the field of accelero-summable transseries, and let $H$ be a Hardy field. In real accelero-summation, use the organic average whenever singularities occur on the positive real axis. Accelero-summation conjecture. This process yields a composition-preserving $H$-field isomorphism

$$\mathbb{T}^{\operatorname{da}}\longrightarrow H$$

onto a Hardy field contained in $\mathcal C^{\omega}$. This would extend the transseries-to-function correspondence while preserving composition, a property not obtained by adjoining solutions one at a time; the source gives no resolution of the conjecture.