It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to $Col(\kappa,\mathbb{P})$, the collection of partial functions from $\kappa$ to $\mathbb{P}$ of size $<\kappa$, ordered by inclusion. Hence if $\mathbb{P}$ is any $\kappa$-closed forcing, then $\mathbb{P} \times Col(\kappa,\mathbb{P}) \sim Col(\kappa,\mathbb{P})$, so $\mathbb{P}$ is completely embeddable into $\mathcal{B}(Col(\kappa,\mathbb{P}))$.

If $\mathbb{P}$ is $\kappa$-strategically closed (following the notation from Cummings' chapter of the *Handbook of Set Theory*, not Jech's book), then one can show that $\mathbb{P}$ is completely embeddable into $\mathcal{B}(Col(\kappa,\mathcal{P}(\mathbb{P})))$, since after forcing with this collapse we can use the strategy to build a filter on $\mathbb{P}$ that is generic over the ground model. It is not clear to me whether we can always embed such $\mathbb{P}$ into the smaller forcing $Col(\kappa,\mathbb{P})$.

**Question:** Are there counterexamples to the statement, "If $\kappa$ is regular and $\mathbb{P}$ is $\kappa$-strategically closed, then there is a complete embedding of $\mathbb{P}$ into $\mathcal{B}(Col(\kappa,\mathbb{P}))$?