Strength of the open-sets-of-reals coding principle relative to the bootstrap/projection principle

Determine whether, over the base theory RCA^ω_0 (the higher-order reverse mathematics base system), the principle that every open set O ⊂ ℝ admits a second-order RM code (as in Simpson’s II.5.6) is strictly weaker than the bootstrap/projection principle $$, i.e., whether the open-sets-of-reals coding principle fails to imply $$. Here $$ asserts: for every type-2 functional Y there exists a subset X ⊂ ℕ such that for all n ∈ ℕ, n ∈ X if and only if there exists f ∈ ℕ^ℕ with Y(f,n)=0.

Background

Section 4 introduces second-countability notions for metric spaces and formulates Principle [$_{0}$], which asserts that for any sequentially compact metric space (M,d) with M ⊂ ℕ^ℕ, every open set can be expressed as a countable union of open balls. The authors note an associated principle for open sets of reals, previously studied in their earlier work.

Feferman’s projection (or bootstrap) principle $$is a strong third-order axiom in this paper’s framework; it can yield powerful consequences such as various convergence theorems and higher-level comprehension. The question asks for the exact logical strength of the reals version of the open-set coding principle relative to$$.

Clarifying this relationship would refine the hierarchy of principles that code topological constructs in reverse mathematics and illuminate how far one can go using second-countability or coding assumptions without invoking stronger non-constructive axioms.