Deriving enumeration of strongly countable sets from choice for closed sets of positive measure

Investigate whether the principle that for every sequence (C_n) of non-empty closed subsets of [0,1] with each C_n of positive Lebesgue measure there exists a sequence (x_n) with x_n ∈ C_n for all n, implies the principle that every strongly countable subset A ⊆ [0,1] can be enumerated (i.e., whether $_{1}$ follows from choice restricted to closed sets of positive measure).

Background

The paper studies various choice-like principles for closed sets under different codings (RM-codes and separably closed-set codes), relating them to systems in hyperarithmetical analysis.

The authors prove implications under certain codings but explicitly note they are not able to derive the strongly countable enumeration principle $_{1}$ from the version of choice restricted to closed sets of positive measure, suggesting a potential separation dependent on coding.

Clarifying this derivability would sharpen the understanding of how measure and coding of closed sets influence the strength of associated choice principles.