Axiomatizability of existentially closed globally valued fields

Let $e$ be a real number. A globally valued field has archimedean error $e$ when its archimedean valuation data satisfy the error parameter specified by $e$. A globally valued field is existentially closed if, for every extension $M\to N$ of globally valued fields, every quantifier-free formula $f$ with parameters $a\in M^V$, and every bound $r\in\mathbf R_+$, an approximate solution in $N$ of height at most $r$ can be approximated in $M$ with height at most $r+\eps$ for every $\eps>0$.

Axiomatizability conjecture. There exists a theory in the language of globally valued fields whose models are exactly the existentially closed globally valued fields of archimedean error $e$.

For ordinary algebraically closed and real closed fields, analogous characterizations are provided by the Nullstellensatz and Positivstellensatz. The corresponding first-order axiomatizability question for globally valued fields is presented as open; the claim concerns whether the existentially closed fields with fixed archimedean error admit such an axiomatization.