Lindelöf property of the Vietoris power V(ω^ω)

Determine whether the Vietoris power space V(ω^ω) (the space of all functions from ω to ω endowed with the Vietoris power topology generated by basic sets of the form [U; Λ, V] with U ⊆ ω, finite Λ ⊆ ω, and V:Λ→𝒫(ω)) is Lindelöf; that is, whether every open cover of V(ω^ω) has a countable subcover.

Background

The paper develops the Vietoris power topology and analyzes many properties of V(ω^ω), establishing that it is separable, first-countable, zero-dimensional, Baire, and has weight and spread equal to 𝔠. However, several basic covering and separation properties are left unresolved. In Table 1 the authors mark certain entries with “?” and then formalize these unknowns as Question \ref{question:VBaire}. Deciding Lindelöfness would clarify how covering properties behave for Vietoris powers over discrete countable bases.