Unbounded positive existential rank for global fields

Let $K$ be a global field and let $\mathscr{G}$ be a finite set of field generators. Consider $K$ as a structure over $\mathscr{L}=\mathscr{L}_a\cup\mathscr{G}$. Global-field rank conjecture. The structure $K$ has unbounded positive existential rank. In particular, it does not have positive existential $r$-catalogues for any $r\geq 1$. This extends the known bounded-rank behavior of $\mathbb{N}$ and $\mathbb{Z}$ to a conjectural contrast for global fields, and would rule out positive existential catalogues over such fields.