The 2-tactic conjecture for the strongly monotonic game

Let $\kappa$ be an infinite cardinal. In the game $SMG([\kappa]^{<\aleph_{0}})$, player ONE plays an increasing sequence of members of the generated ideal and TWO responds with members of $[\kappa]^{<\aleph_{0}}$; a winning 2-tactic is a winning strategy for TWO depending only on the last two moves of ONE. The 2-tactic conjecture. Player TWO has a winning 2-tactic in the game

$$SMG([\kappa]^{<\aleph_{0}})$$

for each infinite cardinal number $\kappa$. The conjecture is motivated by consistency results establishing analogous 2-tactic conclusions for broad classes of cardinals and by the known 3-tactic result for the very strong game. It seeks a ZFC result without additional set-theoretic hypotheses.