Integer parts of real closed fields

Tennenbaum property

Je\v{z}\'abek and Ko{\l}odziejczyk show [1] that if $A$ is a model of Open Induction and the real closure of $A$ is recursively saturated, then $+^A$ and $\leq^A$ cannot be both recursive, and they ask whether it is true that if every unbounded real closed field with an integer part satisfying an arithmetic theory $T$ is recursively saturated, then $T$ has no recursive nonstandard models.