P1-win conjecture for infinitely many disjoint complete boards

Let $G$ be the target graph from the surrounding discussion, and let $\bigsqcup_\omega K_n$ denote a countable disjoint union of copies of the complete graph $K_n$. In the strong Ramsey game $\mathcal{R}(B,G)$, players alternately claim previously unclaimed edges of $B$, and the first player to claim a copy of $G$ wins; if neither does so in finite time, the game is a draw. P1-win conjecture. For $n$ sufficiently large, the game

$$\mathcal{R}(\bigsqcup_\omega K_n,G)$$

is a P1-win. This is presented as a natural question related to the behaviour of Ramsey games on infinite disjoint unions; the source gives no resolution.