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

It was proved recently that Telgarsky's conjecture, which concerns partial information strategies in theBanach-Mazur game, fails in models ofGCH+square. The proof introduces a combinatorial principle that is shown to follow from GCH + square, namely: del: Every separative poset P with the kappa-cc contains a dense sub-poset D such that |{q is an element of D : p extends q}| < kappa for every p is an element of P. We prove this principle is independent of GCH and CH, in the sense that del does not imply CH, and GCH does not imply del assuming the consistency of a huge cardinal. We also consider the more specific question of whether del holds with P equal to the weight-N-omega measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of ZFC + GCH.