Global co-stationarity of the ground model from a new countable length sequence

Suppose $V\subseteq W$ are models of ZFC with the same ordinals, and that for all regular cardinals $\kappa$ in $W$, $V$ satisfies $\square_{\kappa}$. If $W\setminus V$ contains a sequence $r:\omega\rightarrow\gamma$ for some ordinal $\gamma$, then for all cardinals $\kappa<\lambda$ in $W$ with $\kappa$ regular in $W$ and $\lambda\ge\gamma$, $(\mathscr{P}_{\kappa}(\lambda))^W\setminus V$ is stationary in $(\mathscr{P}_{\kappa}(\lambda))^W$. That is, a new $\omega$-sequence achieves global co-stationarity of the ground model.