 Solution

Player II can force both parts of II's winning condition. First choose one coordinate $n$ reserved outside the coding of a fixed infinite descending chain and play $y_n>x_n$. On the remaining reserved coordinates, play values that make the relation coded by $y$ contain that descending chain. Thus $y\notin \mathrm{WF}$ while $y_n\nleqq x_n$ for the chosen $n$. Hence