Embedding the prime model of real exponentiation into o-minimal exponential fields

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o-minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o-minimal exponential field, where the embedding is not necessarily elementary. This is a consequence of an unconditional model theoretic embeddability result that we obtain by applying K\H{o}nig's Lemma.