Does every real-closed field satisfy the first-order least upper bound axiom schema?

Question

The theory of real-closed fields is usually axiomatized by the axioms of ordered fields, along with an axiom for existence of square roots of positive elements, and also an axiom schema stating that every odd degree polynomial has a root. I wonder, do these axioms imply the first-order least upper bound axiom schema? After all, every real-closed field is elementarily equivalent to the ordered field of real numbers, and certainly the real numbers satisfy the least upper bound axiom schema. If the answer is yes, can someone sketch a proof that the axioms of real-closed fields imply the first-order least upper bound axiom schema?