The refined $\mathbb{T}$-conjecture for existentially closed $H$-fields

Let $\mathbb{T}$ be the differential field of transseries, and let an $H$-field with small derivation be an ordered differential field satisfying the relevant $H$-field axioms and smallness condition. Let $\mathcal L$ be the language of ordered valued differential rings. The refined $\mathbb{T}$-conjecture. The field $\mathbb{T}$ is an existentially closed $H$-field, and there exists a set $\Sigma$ of $\mathcal L$-sentences such that the existentially closed $H$-fields with small derivation are exactly the $H$-fields satisfying $\Sigma$. Equivalently, the theory of $H$-fields with small derivation has a model companion, and $\mathbb{T}$ is a model of this model companion. This refines model completeness by identifying the intended existentially closed models; the supplied text gives no resolution.