Is there a unitary $U_{AB}$ such that, for any density operator $\rho$, we have $${\rm {Tr}}_A [U_{AB} (\frac{I_A}{2} \otimes \rho_B)U_{AB}^{\dagger}]= \frac{\rho_B}{2}+\frac{I_B}{4}$$ $${\rm {Tr}}_B [U_{AB} (\frac{I_A}{2} \otimes \rho_B)U_{AB}^{\dagger}]= \frac{\rho_A}{2}+\frac{I_A}{4},$$ where $I_A=I_B=I$ is the identity matrix, $\rho_A=\rho_B=\rho$, A and B are both single qubit systems.
I have thought about decomposing $U_{AB}$ to express the partial trace but failed. I also considered about searching for such unitary, but $U_{AB}$ and $\rho$ are both unknown, which holds me back. Any ideas or comments, both in the analytic way or a computation way, would be appreciated.
It seems that symmetry can be used to derive the proof.