Menger property of 𝔽(ℝ,ord)

Determine whether the space 𝔽(ℝ,ord)—the subspace of the Vietoris power V(ℝ^ω) consisting of functions from ω to ℝ with finite range—is Menger, i.e., whether it satisfies the selection principle S_fin(𝒪, 𝒪).

Background

The authors point out that resolving the Menger property of 𝔽(ℝ,ord) would impact the broader equivalence question between G_fin(ΩX,Ω_X) and G_fin(𝒪{𝔽(X,ord)},𝒪_{𝔽(X,ord)}). Since ℝ is a classical ground space, understanding whether its finite-range Vietoris power is Menger would clarify how covering properties transfer to ordered Vietoris subspaces.