Orthogonal Subspace: $A_p$ and $B_p$ are orthogonally projected by the same projector $P$. Is there always $P$ so that $A_p=QB_p$, where $Q\in O(k)$?

Question

Orthogonal Subspace: $A_p$ and $B_p$ are orthogonally projected by the same projector $P$. Existence of $P$ always so that $A_p=QB_p$, where $Q\in O(k)$?

Let $A,B\in \mathbb{R}^{k\times n}$ with $A\ne B$ and $n\ge k$. Let the matrices have at least $k$ linearly independent columns i.e. $\operatorname{rank}(A)=\operatorname{rank}(B)=k$.

Let $P\in \mathbb{R}^{n\times n}$ be an orthogonal projection matrix with a rank of $k$.

The $k\times n$ matrices $A_p = AP$ and $B_p = BP$ are respectively the projections of $A$ and $B$ onto the row space of $P$. Does the $P$ always exist so that $A_p=QB_p$, where $Q$ is orthogonal?

Answer 1

No. Consider the special case where the matrices are square, $B=P=I$ and $A=2B$.