Does there exist a constructive proof that the arithmetic mean strictly exceeds the geometric mean for all distinct positive reals, without invoking the law of excluded middle or dependent choice?

The classical AM-GM inequality states that the arithmetic mean of distinct positive real numbers strictly exceeds their geometric mean, and this is provable in standard mathematics. However, in constructive mathematics, where every proof must yield an explicit computational witness, many classical inequalities split into weaker or inequivalent forms. The open problem is whether the strict version of AM-GM can be established in a fully constructive setting, meaning within Bishop-style constructive mathematics or a comparable framework, using only constructively valid logical principles and without any appeal to decidability assumptions about real number comparisons.