The independence of GCH and a combinatorial principle related to Banach–Mazur games

It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of GCH+ □. The proof introduces a combinatorial principle that is shown to follow from GCH+ □ , namely: ▽ :Every separative poset P with the κ-cc contains a dense sub-poset D such that |{q∈D:pextendsq}|<κ for every p∈ P. We prove this principle is independent of GCH and CH, in the sense that ▽ does not imply CH, and GCH does not imply ▽ assuming the consistency of a huge cardinal. We also consider the more specific question of whether ▽ holds with P equal to the weight-ℵ_ω measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of ZFC+ GCH.