Here I consider only the case when the matrices are real. Let $C_n$ be the best bound $c$ in dimension $n$. I did not write the entire proof, but it is clear (for me) that $C_2=\sqrt{2}$. Morover $||I-V||=||I-U||$ if $U$ is the unitary matrix associated to $AB$ and $V$ to $diag(A,1)diag(B,1)$. Then $(C_n)_n$ is non-decreasing. Numerical experiments show that $C_3>1.549,C_4>1.564,C_5>1.678,C_6>1.721$.

The interesting question is: Is $\lim_n C_n=2$ true ?

Here I consider only the case when the matrices are real. Let $C_n$ be the best bound $c$ in dimension $n$. I did not write the entire proof, but it is clear (for me) that $C_2=\sqrt{2}$. Morover $||I-V||=||I-U||$ if $U$ is the orthogonal matrix associated to $AB$ and $V$ to $diag(A,1)diag(B,1)$. Then $(C_n)_n$ is non-decreasing. Numerical experiments show that $C_3>1.549,C_4>1.564,C_5>1.678,C_6>1.721$.

The interesting question is: Is $\lim_n C_n=2$ true ?