Is this possible to decompose 2 arbitrary unitary matrices as: $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$?

Question

I guess this statement is true, but I can't prove it:

"For any 2 arbitrary unitary matrices $U_{1}$ and $U_{2}$, we can always decompose them into $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$, where $A$ and $B$ are unitary, and $D$ is diagonal."

Please prove or disprove it. It's good enough to discuss this statement by $2\times2$ matrices.

And the discussion with any of the following constraints is also helpful:

1. $U_{1}\neq U_{2}$;
2. $U_{1}$ and $U_{2}$ are nondegenerate;
3. $U_{1}$ and $U_{2}$ have no common eigenvalue;

Answer 1

As suggested by a comment, you may first unitarily diagonalise $U_1U_2$ as $A D^2 A^\dagger$. Since $U_1U_2$ is a unitary matrix, the eigenvalue matrix $D^2$ is unitary too. Now define $B = U_1^\dagger AD$, where $D$ is an entrywise square root of $D^2$. Then we are done.