Remark: The information is not deleted from the full dilation

At $\lambda =0$, $u_0$ and $v_0$ are orthogonal. For the Bell input, the complete output has the form

The accessible reference state is only classically correlated. The retained $RF$ isometry still preserves the input qubit. Discarding $F$ creates the accessible dephasing channel. Keeping only the original vacuum residual $E$ would incorrectly hide the environmental information.