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Suppose $A$ and $B$ are square Boolean matrices. If $\hat{P} (A \otimes A) = (B \otimes B) \hat{P}$ for some permutation matrix $\hat{P}$, does it follow that $\hat{P} = (P \otimes P)$ for some permutation matrix $P$? In other words, can the matrix witnessing permutation-similarity in the higher dimension be "factored" into a matrix witnessing permutation-similarity in the lower dimension?

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The answer is no. This fails drastically in the case that $A = B = I$, since the equality holds for any matrix $\hat P$ (so it isn't necessarily true that $\hat P = P \otimes P$ for some $P$).

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