Telgarsky's separation conjecture for Banach–Mazur k-tactics

Let $k$ be a positive integer. A winning $k$-tactic for TWO is a strategy in which TWO's move depends only on the most recent $k$ moves of ONE. For a topological space $(X_k,\tau_k)$, let $BM(X_k,\tau_k)$ denote the Banach–Mazur game in which the players choose nonempty open sets satisfying

$$O_{n+1}\subseteq T_n\subseteq O_n.$$

Telgarsky's conjecture. For each positive integer $k$ there is a topological space $(X_k,\tau_k)$ such that TWO does not have a winning $k$-tactic, but does have a winning $k+1$-tactic in the game $BM(X_k,\tau_k)$. This conjecture asks whether the hierarchy of finite-memory tactics in the Banach–Mazur game is strict at every level. The paper notes that known examples do not settle the conjecture; in particular, its results eliminate one candidate example by showing that TWO has a winning $2$-tactic there.