Two very different stability problems
A lyophilized peptide and a reconstituted one are effectively different materials from a stability standpoint. In the dry state, molecular mobility is low, water is largely absent, and the reactions that dominate in solution slow by orders of magnitude. Sealed dry material tolerates ambient temperature for extended periods.
In solution, everything accelerates. Hydrolysis of the backbone, deamidation of asparagine and glutamine, oxidation of methionine and tryptophan, aggregation and adsorption to container surfaces all proceed at rates that make refrigeration and short working windows genuinely necessary.
Most confusion about peptide storage comes from applying solution-phase intuition to dry material or the reverse. Separating the two cases resolves nearly every practical question, including the recurring one about whether a package needed to arrive cold.