Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$.
In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and L.v.d. Dries proved that $No(\kappa)$ is isomorphic to $\mathbb{R}((x))^{No(\kappa)}_{<\kappa}$ which is the subfield of Hahn series of length $< \kappa$ over $\mathbb{R}$ with value group $No(\kappa)$.
Let $S$ be the subset of $\mathbb{R}((x))^{No(\kappa)} = \mathbb{R}((x))^{No(\kappa)}_{<{\kappa}^+}$ of Hahn series of either length $<\kappa$ or of length $\kappa$ whose $\kappa$-sequence of exponents is cofinal in $No(\kappa)$.
It is not too difficult to see that $S$ is stable under $+$ and $-$.
I wonder if it is a subfield of $\mathbb{R}((x))^{No(\kappa)}$, in which case it would be an example of completion of $No(\kappa)$. Does anyone know how to prove/disprove this?