Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x$ and $y$ there are integers $n$ and $m$ such that $x^n>y$ and $y^m>x$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?