Restatement (Second) Contracts § 87(1)(a) specifies that the promise must be in writing and signed by the offeror. It is important not to overgeneralize here. Rex's promise is not enforceable under the rule specified in § 87(1)(a), but it does not follow that it is unenforceable. All that follows is that the rule in § 87(1)(a) does not make it enforceable.
§ 87(1)(a) does not require that the promise be in a signed writing to be enforceable. The section just states a sufficient, not a necessary condition for enforceability: The promise to hold the offer open for a specified time, is legally enforceable if (1) the promise is in writing; (2) signed by the offeror; (3) the writing recites the consideration given in exchange for the promise; and, (4) the bargain proposed by the offer involves a fair exchange in a reasonable time period. That leaves open the question of whether an oral promise may, under appropriate conditions, be enforeceable as an option contract.