The independence of $$\mathsf {GCH}$$ GCH and a combinatorial principle related to Banach–Mazur games

It was proved recently that Telgarsky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of $$\mathsf {GCH}+\square $$ . The proof introduces a combinatorial principle that is shown to follow from $$\mathsf {GCH}+\square $$ , namely: We prove this principle is independent of $$\mathsf {GCH}$$ and $$\mathsf {CH}$$ , in the sense that $$\bigtriangledown $$ does not imply $$\mathsf {CH}$$ , and $$\mathsf {GCH}$$ does not imply $$\bigtriangledown $$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $$\bigtriangledown $$ holds with $${\mathbb {P}}$$ equal to the weight- $$\aleph _\omega $$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $$\mathsf {ZFC}+\mathsf {GCH}$$ .