  It was proved recently that Telg\'arsky'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:
  $\triangledown$: Every separative poset $\mathbb P$ with the $\kappa$-cc
contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p
\text{ extends } q \}| < \kappa$ for every $p \in \mathbb P$.
  We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$,
in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and
$\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a
huge cardinal.
  We also consider the more specific question of whether $\triangledown$ 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 $\mathsf{ZFC}+\mathsf{GCH}$.
