The $2$-co-lacunary subsequence principle for arbitrary noncommutative $L_1$-spaces

For every von Neumann algebra $\text{M}$, we prove that each bounded sequence in the Haagerup space $L_1(\text{M})={\text{M}}_{\ast }$ admits either a norm-convergent subsequence or a $2$-co-lacunary subsequence. This answers an open problem of Parcet and Randrianantoanina \cite[Remark~3.11]{ParcetRandrianantoanina2006} and removes both hyperfiniteness and semifiniteness.