The one-live-qubit obstruction

Let ${\omega }_{\mathcal{D}}$ be the normalized Choi state of $\mathcal{D}$, ordered as output $0$ and input-reference positions $1,2,3$. From (eq:reps),

Set

They commute and $S_1{S}_3=Y_0{X}_1{Y}_2{X}_3$. Partial transpose on position 1 therefore gives

On the simultaneous eigenspace $S_1=-1,S_3=+1$, its eigenvalue is $-1/8$. That eigenspace is nonzero because the two commuting Paulis are independent.