Telgarsky's conjecture may fail

Telg\'arsky's conjecture states that for each $k \in \mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\mathsf{GCH}+\square$, that if {\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $\pi$-base with certain nice properties, then {\scriptsize NONEMPTY} is able to encode, in each consecutive pair of her opponent's moves, all essential information about the play of the game before the current move. Our proof shows that under $\mathsf{GCH}+\square$, every $T_3$ space has a sufficiently nice $\pi$-base that enables this coding strategy. Translated into the language of partially ordered sets, what we really show is that $\mathsf{GCH}+\square$ implies the following statement, which is equivalent to the existence of the "nice'' $\pi$-bases mentioned above: \emph{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 that this statement is independent of $\mathsf{ZFC}$: while it holds under $\mathsf{GCH}+\square$, it is false even for ccc posets if $\mathfrak{b} > \aleph_1$. We also show that if $|\mathbb P| < \aleph_\omega$, then \axiom-for-$\mathbb P$ is a consequence of $\mathsf{GCH}$ holding below $|\mathbb P|$.