The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when the sequence has a lower bound.

A strategy is a function $\sigma : \mathbb P^{<\omega} \to \mathbb P$ such that for each linearly ordered $\vec x \in \mathbb P^{<\omega}$, $\sigma(\vec x) \leq \min(\vec x)$.

A tactic is a decreasing function $\tau : \mathbb P \to \mathbb P$.

The Banach-Mazur game on $\mathbb P$ is $\omega$-strategically closed when there is a strategy $\sigma$ such that II wins whenever II plays according to $\sigma$, meaning if the sequence of plays so far is $\vec x$ and it is II’s turn, then II plays $\sigma(\vec x)$. The game is $\omega$-tactically-closed when II has a winning tactic, meaning II wins when they always play $\sigma(a)$, where $a$ is the last move by I.

These notions have generalizations to games of length longer than $\omega$. The separation of tactical and strategic closure for games of uncountable length has been studied by Yoshinobu. The situation is different because the games involve limit stages. However, I don't know whether these notions have been separated for games of length $\omega$.

Question: Suppose $\mathbb P$ is $\omega$-strategically-closed. Is it $\omega$-tactically-closed?